I argue that, despite repeated claims of Ibata et al., the globular cluster
NGC 2419 does not pose a problem for modified Newtonian dynamics (MOND). I
present a new polytropic model with a running polytropic index. This model
provides an improved representation of the radial distribution of surface
brightness while maintaining a reasonable fit to the velocity dispersion
profile. Although it may be argued that the differences with these observations
remain large compared to the reported random errors, there are several
undetectable systematic effects which render a formal likelihood analysis
irrelevant. I comment generally upon these effects and upon the intrinsic
limitations of pressure supported objects as tests of gravity.Comment: 3 page, 2 figure

Baumbardt

Ibata

R. H. Sanders

Sanders

Scarpa

English

arXiv

Sanders, R. H.

University of Groningen Research Database

  
 University of Groningen
NGC 2419 does not challenge MOND, Part 2
Sanders, R. H.
Published in:
Monthly Notices of the Royal Astronomical Society
DOI:
10.1111/j.1745-3933.2012.01227.x
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Astronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.x
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Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.x
NGC 2419 does not challenge MOND, Part 2
R. H. Sanders
Kapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the Netherlands
Accepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21
ABSTRACT
I argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does not
pose a problem for modified Newtonian dynamics (MOND). I present a new polytropic model
with a running polytropic index. This model provides an improved representation of the radial
distribution of surface brightness while maintaining a reasonable fit to the velocity dispersion
profile. Although it may be argued that the differences with these observations remain large
compared to the reported random errors, there are several undetectable systematic effects
which render a formal likelihood analysis irrelevant. I comment generally upon these effects
and upon the intrinsic limitations of pressure-supported objects as tests of gravity.
Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.
1 I N T RO D U C T I O N
Ibata et al. (2011a, henceforth I1) have claimed that observations of
the distant globular cluster NGC 2419 constitute a severe challenge
for modified Newtonian dynamics (MOND), and that this object is,
in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-
marily, upon Newtonian and MONDian Michie models in which the
phase space distribution is assumed to be of specific form, leading
to a density cut-off at a finite radius. In my initial response to their
paper, I argued that these models were too restricted and presented
high n polytropic MOND models (Sanders 2011, henceforth S11),
in which the radial velocity dispersion decreases with radius; to the
eye in any case, such models appear to be quite consistent with both
the observed radial distribution of surface brightness and the radial
profile of velocity dispersion in this cluster. Furthermore, I cau-
tioned that it is questionable to rely on formal errors in likelihood
analyses of observations which may well contain observational or
intrinsic systematic effects.
Ibata et al. (2011b, henceforth I2) responded with their own
analysis of polytropes and assert that they had already, in effect,
considered, and ruled out, a range of non-isothermal MOND mod-
els which fitted the data better than the polytropic model that I
presented; that, in any case, Newtonian Michie models fit the obser-
vations significantly better than the MOND non-isothermal models.
Again, this conclusion is based upon the use of random errors in a
likelihood analysis. It is primarily this point that I wish to address
here because it has general relevance to the interpretation of astro-
nomical observations. But first, I present a new model in which the
polytropic index runs with radius.
2 M O D E L A N D D I S C U S S I O N
I2 point to the problem for polytropic MOND models as a tension
between the predicted surface density distribution and the velocity
E-mail: sanders@astro.rug.nl
dispersion radial profile. This can be summarized, in my own words,
as follows: to more precisely fit the fall-off of surface density within
the very small formal errors, in particular near the outer cut-off, the
polytropic index must be rather large (n > 15); that is, the system
should be closer to isothermal. But to fit the radial decrease in the
line-of-sight velocity dispersion, the index should be smaller (n ≈
10). This result is fairly insensitive to the anisotropy radius – the
distance beyond which radial orbits begin to dominate.
One obvious solution to this problem is to allow the polytropic
index to increase with radius (there is nothing sacred about the rigid
polytropic pressure–density relation). Therefore, I have considered
several prescriptions in which n increases beyond 100 pc or so, for
example,
n = n0 exp[r/rp]. (1)
Below, I show the results for one such model. Fig. 1 is the observed
surface density of stars as a function of radius (points) and that
predicted by the model (solid curve), and Fig. 2 is the same for the
line-of-sight velocity dispersion. For this model, the central radial
velocity dispersion is taken to be 7.7 km s−1, the central density is
38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropic
index is 9.2 and the scale length for the growth of the polytropic
index (rp) is 850 pc. Note that with a variable polytropic index, the
structure equation (Jeans equation) for the run of density contains
an additional term reflecting the gradient in n. The total mass of this
model is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. The
value of the MOND acceleration parameter is a0 = 10−8 cm s−2 and
the MOND interpolation function was taken to be standard form as
in S11.
Now, is this an acceptable match to the observations? It is clearly
an improvement over my initial polytropic MOND model with a
fixed n, particularly in matching the surface density distribution
while maintaining a reasonable representation of the velocity dis-
persion profile. But Ibata et al. will probably tell us, based upon
a maximum likelihood analysis, that it is completely ruled out by
the observations or that it is 138 times less probable than the best
C© 2012 The Author
Monthly Notices of the Royal Astronomical Society C© 2012 RAS
D
ow
nloaded from
 https://academ
ic.oup.com
/m
nrasl/article-abstract/422/1/L21/971080 by guest on 21 N
ovem
ber 2018
L22 R. H. Sanders
Figure 1. Surface brightness distribution in the globular cluster NGC 2419
(points) compared to the MOND non-isothermal (polytropic with running
index), non-isotropic model described in the text.
Figure 2. The observed radial dependence of the line-of-sight velocity
dispersion in NGC 2419 (points) compared to that of the MOND model
shown above.
Newtonian Michie model. Such claims, of course, rest on the fact
that the differences between model and observations, while small to
the eye, are still larger than the reported measurement errors (com-
parable to the size of the points in the surface density distribution).
But these statements are misleading because they give an impres-
sion of precision that is not, and cannot be, present in astronomical
observations of one such object.
I2 point out that in spherical symmetry, given the anisotropy
factor of a model, β(r), there is a unique relationship between the
run of radial velocity and the density distribution. This is certainly
true, but the observed quantities are not the radial velocity or the
density but the projected versions of these quantities. We know
very well that small errors in the surface density can translate into
significant differences in the true density and, hence, in the line-
of-sight velocity dispersion (I2 admit that there are problems with
de-projection particularly in the outer regions where the errors are
larger).
But let us assume that I1 and I2 perfectly understand the sys-
tematic and random errors in their observations, and that these are
vanishingly small. Then there remains a number of ‘known un-
knowns – or more precisely ‘known unknowables’, for example,
the symmetry of the cluster. I1 and I2 argue that the cluster appears
to be quite round and that this supports the assumption of spherical
symmetry. But if the cluster is oblate or prolate with the symmetry
axis lying near the line of sight, then the cluster would still appear to
be round without being spherically symmetric. This would clearly
alter the relation between the run of line-of-sight velocity disper-
sion and the projected density distribution, easily by more than the
random measurement errors.
Then there is the unknown of rotation. I1 consider an earlier
specific claim of rotation and dismiss this as an effect of small
number statistics. But they cannot dismiss the possibility of rotation
in general. Rotation is most likely to be present in the outer regions
where a systematic velocity of about 4 km s−1 would be sufficient
to reduce the model velocity dispersion at the outermost measured
point in Fig. 2 to the observed value. If the rotation axis were
within 30◦ of the line of sight, its projection would be less than 2.0
km s−1; given the lower surface density of stars near the cut-off, it
is difficult to believe that this would be detectable even in the very
careful observations of I1.
The point is that conceivable but undetectable intrinsic systematic
effects vitiate the value of likelihood analyses (I have not even
mentioned the ‘unknown unknowns’ – effects that we have not
thought of). If so, it would not be the first time that sophisticated
statistical analyses have been applied to astronomical observations
dominated by systematic effects. While Ibata et al. are certainly
aware of these various caveats (I1), they continue (I2) to place an
unrealistically high value on ‘the objective tool that is statistical
inference’.
There are fewer ambiguities with galaxy rotation curves. This is
essentially because a two-dimensional object (presumably a highly
flattened disc) is being projected on to a two-dimensional sky. There
is one component of velocity and the vector most often lies in
one plane. The optical appearance of the disc combined with a
two-dimensional radial velocity field can generally unambiguously
determine the projection, or at least reveal when problems – such
as non-circular discs, non-planar motion, warping or non-circular
motion – are present. As is well known, MOND works extremely
well in predicting the shapes of rotation curves using the observed
distribution of baryonic matter (Sanders & McGaugh 2002). But
even so, the rotation curves in a few per cent of these well-specified
systems are not in agreement with those predicted by MOND, nor
would we expect them to be. A perfect theory would not precisely
predict the rotation curves of all disc galaxies because of the intrinsic
uncertainties.
It is worse for spheroidal pressure-supported objects because of
uncertainties due to projection – shape, rotation – as well as the
degree and radial dependence of anisotropy of the velocity field
and, in relaxed systems such as globular clusters, the possibility of
mass segregation and varying mass-to-light ratio. If Ibata et al. had
several such objects, all showing the same discrepancies, their case
would be stronger, but with only one cluster, the small differences
between model predictions and observations are hardly definitive. I
point out that opposite conclusion has been reached by Scarpa et al.
(2011) based upon observations of distant globular clusters which
appear to be inconsistent with Newton (see also the more general
discussion of Baumbardt, Grebel & Kroupa 2005). The point is
that strong statements on the nature of gravity cannot be made
from observations of a single such cluster given the uncertainties
intrinsic in astronomical observations of a three-dimensional object
projected on to two dimensions.
AC K N OW L E D G M E N T S
I am grateful to Moti Milgrom for helpful comments on a draft
of this Letter, and I thank an anonymous referee of my previous
C© 2012 The Author, MNRAS 422, L21–L23
Monthly Notices of the Royal Astronomical Society C© 2012 RAS
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nloaded from
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/m
nrasl/article-abstract/422/1/L21/971080 by guest on 21 N
ovem
ber 2018
NGC 2419 does not challenge MOND, Part 2 L23
paper on this subject for suggesting the idea of a running polytropic
index.
R E F E R E N C E S
Baumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1
Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., Delessandro
E., 2011a, ApJ, 738, 186 (I1)
Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., Dalessandro
E., 2011b, ApJ, 743, 43 (I2)
Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x
(S11)
Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263
Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,
525, 148
This paper has been typeset from a TEX/LATEX file prepared by the author.
C© 2012 The Author, MNRAS 422, L21–L23
Monthly Notices of the Royal Astronomical Society C© 2012 RAS
D
ow
nloaded from
 https://academ
ic.oup.com
/m
nrasl/article-abstract/422/1/L21/971080 by guest on 21 N
ovem
ber 2018


NGC 2419 does not challenge MOND, Part 2

University of Groningen

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 26-04-2023Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

I argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does not pose a problem for modified Newtonian dynamics (MOND). I present a new polytropic model with a running polytropic index. This model provides an improved representation of the radial distribution of surface brightness while maintaining a reasonable fit to the velocity dispersion profile. Although it may be argued that the differences with these observations remain large compared to the reported random errors, there are several undetectable systematic effects which render a formal likelihood analysis irrelevant. I comment generally upon these effects and upon the intrinsic limitations of pressure-supported objects as tests of gravity.</p

Proceedings - University of Groningen

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 17-07-2023Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

ARTS repository - University of Groningen

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 07-06-2022Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-10-2022Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

Dissertations of the University of Groningen

   University of GroningenNGC 2419 does not challenge MOND, Part 2Sanders, R. H.Published in:Monthly Notices of the Royal Astronomical SocietyDOI:10.1111/j.1745-3933.2012.01227.xIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2012Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Sanders, R. H. (2012). NGC 2419 does not challenge MOND, Part 2. Monthly Notices of the RoyalAstronomical Society, 422(1), L21-L23. https://doi.org/10.1111/j.1745-3933.2012.01227.xCopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 09-09-2023Mon. Not. R. Astron. Soc. 422, L21–L23 (2012) doi:10.1111/j.1745-3933.2012.01227.xNGC 2419 does not challenge MOND, Part 2R. H. SandersKapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, the NetherlandsAccepted 2012 January 24. Received 2012 January 22; in original form 2011 December 21ABSTRACTI argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does notpose a problem for modified Newtonian dynamics (MOND). I present a new polytropic modelwith a running polytropic index. This model provides an improved representation of the radialdistribution of surface brightness while maintaining a reasonable fit to the velocity dispersionprofile. Although it may be argued that the differences with these observations remain largecompared to the reported random errors, there are several undetectable systematic effectswhich render a formal likelihood analysis irrelevant. I comment generally upon these effectsand upon the intrinsic limitations of pressure-supported objects as tests of gravity.Key words: gravitation – globular clusters: individual: NGC 2419 – dark matter.1 I N T RO D U C T I O NIbata et al. (2011a, henceforth I1) have claimed that observations ofthe distant globular cluster NGC 2419 constitute a severe challengefor modified Newtonian dynamics (MOND), and that this object is,in fact, a ‘crucible’ for theories of gravity. This claim was based, pri-marily, upon Newtonian and MONDian Michie models in which thephase space distribution is assumed to be of specific form, leadingto a density cut-off at a finite radius. In my initial response to theirpaper, I argued that these models were too restricted and presentedhigh n polytropic MOND models (Sanders 2011, henceforth S11),in which the radial velocity dispersion decreases with radius; to theeye in any case, such models appear to be quite consistent with boththe observed radial distribution of surface brightness and the radialprofile of velocity dispersion in this cluster. Furthermore, I cau-tioned that it is questionable to rely on formal errors in likelihoodanalyses of observations which may well contain observational orintrinsic systematic effects.Ibata et al. (2011b, henceforth I2) responded with their ownanalysis of polytropes and assert that they had already, in effect,considered, and ruled out, a range of non-isothermal MOND mod-els which fitted the data better than the polytropic model that Ipresented; that, in any case, Newtonian Michie models fit the obser-vations significantly better than the MOND non-isothermal models.Again, this conclusion is based upon the use of random errors in alikelihood analysis. It is primarily this point that I wish to addresshere because it has general relevance to the interpretation of astro-nomical observations. But first, I present a new model in which thepolytropic index runs with radius.2 M O D E L A N D D I S C U S S I O NI2 point to the problem for polytropic MOND models as a tensionbetween the predicted surface density distribution and the velocityE-mail: sanders@astro.rug.nldispersion radial profile. This can be summarized, in my own words,as follows: to more precisely fit the fall-off of surface density withinthe very small formal errors, in particular near the outer cut-off, thepolytropic index must be rather large (n > 15); that is, the systemshould be closer to isothermal. But to fit the radial decrease in theline-of-sight velocity dispersion, the index should be smaller (n ≈10). This result is fairly insensitive to the anisotropy radius – thedistance beyond which radial orbits begin to dominate.One obvious solution to this problem is to allow the polytropicindex to increase with radius (there is nothing sacred about the rigidpolytropic pressure–density relation). Therefore, I have consideredseveral prescriptions in which n increases beyond 100 pc or so, forexample,n = n0 exp[r/rp]. (1)Below, I show the results for one such model. Fig. 1 is the observedsurface density of stars as a function of radius (points) and thatpredicted by the model (solid curve), and Fig. 2 is the same for theline-of-sight velocity dispersion. For this model, the central radialvelocity dispersion is taken to be 7.7 km s−1, the central density is38 M pc−2, the anisotropy radius is 11.5 pc, the central polytropicindex is 9.2 and the scale length for the growth of the polytropicindex (rp) is 850 pc. Note that with a variable polytropic index, thestructure equation (Jeans equation) for the run of density containsan additional term reflecting the gradient in n. The total mass of thismodel is 6.04 × 105 M, yielding a mass-to-light ratio of 1.4. Thevalue of the MOND acceleration parameter is a0 = 10−8 cm s−2 andthe MOND interpolation function was taken to be standard form asin S11.Now, is this an acceptable match to the observations? It is clearlyan improvement over my initial polytropic MOND model with afixed n, particularly in matching the surface density distributionwhile maintaining a reasonable representation of the velocity dis-persion profile. But Ibata et al. will probably tell us, based upona maximum likelihood analysis, that it is completely ruled out bythe observations or that it is 138 times less probable than the bestC© 2012 The AuthorMonthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018L22 R. H. SandersFigure 1. Surface brightness distribution in the globular cluster NGC 2419(points) compared to the MOND non-isothermal (polytropic with runningindex), non-isotropic model described in the text.Figure 2. The observed radial dependence of the line-of-sight velocitydispersion in NGC 2419 (points) compared to that of the MOND modelshown above.Newtonian Michie model. Such claims, of course, rest on the factthat the differences between model and observations, while small tothe eye, are still larger than the reported measurement errors (com-parable to the size of the points in the surface density distribution).But these statements are misleading because they give an impres-sion of precision that is not, and cannot be, present in astronomicalobservations of one such object.I2 point out that in spherical symmetry, given the anisotropyfactor of a model, β(r), there is a unique relationship between therun of radial velocity and the density distribution. This is certainlytrue, but the observed quantities are not the radial velocity or thedensity but the projected versions of these quantities. We knowvery well that small errors in the surface density can translate intosignificant differences in the true density and, hence, in the line-of-sight velocity dispersion (I2 admit that there are problems withde-projection particularly in the outer regions where the errors arelarger).But let us assume that I1 and I2 perfectly understand the sys-tematic and random errors in their observations, and that these arevanishingly small. Then there remains a number of ‘known un-knowns – or more precisely ‘known unknowables’, for example,the symmetry of the cluster. I1 and I2 argue that the cluster appearsto be quite round and that this supports the assumption of sphericalsymmetry. But if the cluster is oblate or prolate with the symmetryaxis lying near the line of sight, then the cluster would still appear tobe round without being spherically symmetric. This would clearlyalter the relation between the run of line-of-sight velocity disper-sion and the projected density distribution, easily by more than therandom measurement errors.Then there is the unknown of rotation. I1 consider an earlierspecific claim of rotation and dismiss this as an effect of smallnumber statistics. But they cannot dismiss the possibility of rotationin general. Rotation is most likely to be present in the outer regionswhere a systematic velocity of about 4 km s−1 would be sufficientto reduce the model velocity dispersion at the outermost measuredpoint in Fig. 2 to the observed value. If the rotation axis werewithin 30◦ of the line of sight, its projection would be less than 2.0km s−1; given the lower surface density of stars near the cut-off, itis difficult to believe that this would be detectable even in the verycareful observations of I1.The point is that conceivable but undetectable intrinsic systematiceffects vitiate the value of likelihood analyses (I have not evenmentioned the ‘unknown unknowns’ – effects that we have notthought of). If so, it would not be the first time that sophisticatedstatistical analyses have been applied to astronomical observationsdominated by systematic effects. While Ibata et al. are certainlyaware of these various caveats (I1), they continue (I2) to place anunrealistically high value on ‘the objective tool that is statisticalinference’.There are fewer ambiguities with galaxy rotation curves. This isessentially because a two-dimensional object (presumably a highlyflattened disc) is being projected on to a two-dimensional sky. Thereis one component of velocity and the vector most often lies inone plane. The optical appearance of the disc combined with atwo-dimensional radial velocity field can generally unambiguouslydetermine the projection, or at least reveal when problems – suchas non-circular discs, non-planar motion, warping or non-circularmotion – are present. As is well known, MOND works extremelywell in predicting the shapes of rotation curves using the observeddistribution of baryonic matter (Sanders & McGaugh 2002). Buteven so, the rotation curves in a few per cent of these well-specifiedsystems are not in agreement with those predicted by MOND, norwould we expect them to be. A perfect theory would not preciselypredict the rotation curves of all disc galaxies because of the intrinsicuncertainties.It is worse for spheroidal pressure-supported objects because ofuncertainties due to projection – shape, rotation – as well as thedegree and radial dependence of anisotropy of the velocity fieldand, in relaxed systems such as globular clusters, the possibility ofmass segregation and varying mass-to-light ratio. If Ibata et al. hadseveral such objects, all showing the same discrepancies, their casewould be stronger, but with only one cluster, the small differencesbetween model predictions and observations are hardly definitive. Ipoint out that opposite conclusion has been reached by Scarpa et al.(2011) based upon observations of distant globular clusters whichappear to be inconsistent with Newton (see also the more generaldiscussion of Baumbardt, Grebel & Kroupa 2005). The point isthat strong statements on the nature of gravity cannot be madefrom observations of a single such cluster given the uncertaintiesintrinsic in astronomical observations of a three-dimensional objectprojected on to two dimensions.AC K N OW L E D G M E N T SI am grateful to Moti Milgrom for helpful comments on a draftof this Letter, and I thank an anonymous referee of my previousC© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018NGC 2419 does not challenge MOND, Part 2 L23paper on this subject for suggesting the idea of a running polytropicindex.R E F E R E N C E SBaumbardt H., Grebel E. K., Kroupa P., 2005, MNRAS, 259, L1Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DelessandroE., 2011a, ApJ, 738, 186 (I1)Ibata R., Sollima A., Nipoti C., Bellazzini M., Chapman S. C., DalessandroE., 2011b, ApJ, 743, 43 (I2)Sanders R. H., 2011, MNRAS, doi:10.1111/j.1745-3933.2011.01160.x(S11)Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263Scarpa R., Marconi G., Carraro G., Falomo R., Villanova S., 2011, A&A,525, 148This paper has been typeset from a TEX/LATEX file prepared by the author.C© 2012 The Author, MNRAS 422, L21–L23Monthly Notices of the Royal Astronomical Society C© 2012 RASDownloaded from https://academic.oup.com/mnrasl/article-abstract/422/1/L21/971080 by guest on 21 November 2018

I argue that, despite repeated claims of Ibata et al., the globular cluster NGC 2419 does not pose a problem for modified Newtonian dynamics (MOND). I present a new polytropic model with a running polytropic index. This model provides an improved representation of the radial distribution of surface brightness while maintaining a reasonable fit to the velocity dispersion profile. Although it may be argued that the differences with these observations remain large compared to the reported random errors, there are several undetectable systematic effects which render a formal likelihood analysis irrelevant. I comment generally upon these effects and upon the intrinsic limitations of pressure-supported objects as tests of gravity

NGC 2419 does not challenge MOND, Part 2

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