We present calculations illustrating the potential of gravitational
microlensing to discriminate between classical models of stellar surface
brightness profiles and the recently computed ``Next Generation'' models of
Hauschildt et al. These spherically-symmetric models include a much improved
treatment of molecular lines in the outer atmospheres of cool giants -- stars
which are very typical sources in Galactic bulge microlensing events. We show
that the microlensing signatures of intensively monitored point and fold
caustic crossing events are readily able to distinguish between NextGen and the
classical models, provided a photometric accuracy of 0.01 magnitudes is
reached. This accuracy is now routinely achieved by alert networks, and hence
current observations can discriminate between such model atmospheres, providing
a unique insight on stellar photospheres.Comment: 4 pages, 4 figures, Astronomy & Astrophysics (Letters), vol. 388, L1
  (2002

Albrow

Bogdanov

Bontz

Claret

D. Valls-Gabaud

Davis

Gould

H. M. Bryce

Hauschildt

Heyrovský

M. A. Hendry

Nemiroff

Orosz

Simmons

Valls-Gabaud

Van Hamme

Witt

Wittkowski

English

arXiv

Crossref

Gravitational microlensing as a test of stellar model atmospheres

4 pages, 4 figuresWe present calculations illustrating the potential of gravitational microlensing to discriminate between classical models of stellar surface brightness profiles and the recently computed ``Next Generation\'\' models of Hauschildt et al. These spherically-symmetric models include a much improved treatment of molecular lines in the outer atmospheres of cool giants -- stars which are very typical sources in Galactic bulge microlensing events. We show that the microlensing signatures of intensively monitored point and fold caustic crossing events are readily able to distinguish between NextGen and the classical models, provided a photometric accuracy of 0.01 magnitudes is reached. This accuracy is now routinely achieved by alert networks, and hence current observations can discriminate between such model atmospheres, providing a unique insight on stellar photospheres

Bryce, H. M.

Hendry, M. A.

Valls-Gabaud, D.

HAL-INSU

We present calculations illustrating the potential of gravitational microlensing to discriminate between classical models of stellar surface brightness profiles and the computed ""Next Generation"" models of Hauschildt et al. (1999). These spherically-symmetric models include a much improved treatment of molecular lines in the outer atmospheres of cool giants - stars which are very typical sources in Galactic bulge microlensing events. We show that the microlensing signatures of intensively monitored point and fold caustic crossing events are readily able to distinguish between NextGen and the classical models, provided a photometric accuracy of 0.01 mag is reached. This accuracy is now routinely achieved by alert networks, and hence current observations can discriminate between such model atmospheres, providing a unique insight on stellar photospheres

Bryce, H.M.

Hendry, M.A.

Enlighten

       Bryce, H.M., Hendry, M.A. and Valls-Gabaud, D. (2002) Gravitational microlensing as a test of stellar model atmospheres. Astronomy and Astrophysics, 388 (1). L1-L4. ISSN 0004-6361  http://eprints.gla.ac.uk/1397/  Deposited on: 20 February 2012   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk A&A 388, L1{L4 (2002)DOI: 10.1051/0004-6361:20020507c© ESO 2002Astronomy&AstrophysicsGravitational microlensing as a test of stellar model atmospheresH. M. Bryce1;2, M. A. Hendry1, and D. Valls-Gabaud31 Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK2 203 Van Allen Hall, Department of Physics and Astronomy, University of Iowa, Iowa City, IA, 52242, USA3 CNRS UMR 5572, Observatoire Midi-Pyrenees, 14 avenue E. Belin, 31400 Toulouse, FranceReceived 12 December 2001 / Accepted 2 April 2002Abstract. We present calculations illustrating the potential of gravitational microlensing to discriminate betweenclassical models of stellar surface brightness proles and the recently computed \Next Generation" models ofHauschildt et al. These spherically-symmetric models include a much improved treatment of molecular linesin the outer atmospheres of cool giants { stars which are very typical sources in Galactic bulge microlensingevents. We show that the microlensing signatures of intensively monitored point and fold caustic crossing eventsare readily able to distinguish between NextGen and the classical models, provided a photometric accuracyof 0.01 mag is reached. This accuracy is now routinely achieved by alert networks, and hence current observationscan discriminate between such model atmospheres, providing a unique insight on stellar photospheres.Key words. gravitational lensing: stars { stars: atmospheres, fundamental parameters, imaging1. IntroductionRecently gravitational microlensing has been shown tobe an eective tool for stellar astrophysics, primarilythrough high time resolution observations of extendedsource events { i.e. where the lensed star cannot reason-ably be approximated as a point source.The microlensing signatures of extended sources wereinitially discussed in Bontz (1979), Gould (1994), Witt &Mao (1994) and Nemiro & Wickramasinghe (1994) forthe case of a star of uniform surface brightness, wherethe nite size of the star produces a signicant deviationfrom the point source light curve. The primary motivationfor the authors in the 1990’s was to use extended sourceeects to better constrain the properties of the lens { thusdistinguishing between e.g. LMC and galactic lenses.The microlensing signatures of a non-uniform disc wererst considered by Valls-Gabaud (1994), Bogdanov &Cherepashchuk (1995), Witt (1995) and Simmons et al.(1995), who showed that the lightcurves would displaya chromatic signature as the lens eectively sees a starof dierent radius at dierent wavelengths. Valls-Gabaud(1994, 1998) modelled linear, quadratic and logarithmiclimb darkening, with Johnson U to K coecients com-puted using ATLAS LTE model atmospheres from Kurucz(1994), and also predicted a spectroscopic eect, wherethe line proles would change during the microlensingevent, providing another probe of atmospheric structure.Send oprint requests to: e-mail: martin@astro.gla.ac.ukThis was conrmed by Heyrovsky et al. (2000) for a par-ticular red giant model atmosphere.Gould (2001) reviews recent observational progressin detecting non-uniform surface brightness eects. Theclearest evidence of limb darkening to date comes fromanalysis of event MACHO 97-BLG-28 (Albrow et al.1999). This was an example of a binary lens event with acusp crossing { arguably the most favourable type of eventfor stellar astrophysics studies. The photometry and sam-pling coverage for this event were of sucient quality tot a 2-parameter limb darkening law in V and I.Although observations already strongly favour limbdarkened atmospheres over uniform brightness models,the use of microlensing as a discriminant between limbdarkened models is a relatively new subject. Rhie &Bennett (2002) investigate the photometric signaturesof 1- and 2-parameter limb darkening models from foldcaustic crossing events. Albrow et al. (2001) present adetailed analysis of event OGLE 99-BUL-23, comparingthe V and I light curves with several dierent model at-mospheres, including for the rst time a treatment of theerrors in the lens parameters and their correlation with theestimated limb darkening coecients. Notwithstandingthe rigorous statistical approach of that paper, whichraises the benchmark for future analyses, only linear limbdarkening models were considered. The current state ofthe art atmosphere models are the \Next Generation"models of Hauschildt et al. (1999a,b), which include amuch more detailed treatment of molecular absorptionlines in the atmospheres of cool giants and considerLettertotheEditorArticle published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020507L2 H. M. Bryce et al.: Gravitational microlensing as a test of model atmospheresspherically symmetric atmospheres, as opposed to planeparallel models used previously. The aim of this letter is tocompute the microlensing signatures of typical extendedsource events, lensed by point and fold caustics, and toshow that current observations can, in principle, alreadydiscriminate between dierent theoretical models { thusstimulating greater interest in microlensing observationsas a test of stellar model atmospheres.2. PHOENIX Next Generation modelatmospheresComputations of stellar surface brightness proles havebeen carried out for several decades but until very re-cently were based on approximate ts to the limb dark-ening { i.e. the intensity, I(), as a function of (the co-sine of) emergent angle, . For instance a simple linearmodel, namely I() = I0 [1− c1(1− )] was often su-cient, while detailed study of eclipsing binaries sometimesrequired two parameter models, such as the \square root"I() = I01− c2(1− )− c3(1−p)or \logarithmic"model I() = I0 [1− c4(1− )− c5 ln]. Each coe-cient ci depends on the temperature, gravity and chem-ical composition of the source, but the range of validityof these \laws" is often very limited (see Valls-Gabaud1998 and references therein). For instance the square rootformulation provides a good t for hot stars, while aquadratic model ts well for cooler stars. A new non-linearformulation, using 4 parameters and valid across the entirerange of eective temperatures and surface gravities wasrecently suggested by Claret (2000), who also made non-linear ts to the ATLAS model atmospheres, improvingupon the previous calculations of Van Hamme (1993).By contrast, the recent PHOENIX \Next Generation"stellar atmosphere models, computed by Hauschildt et al.(1999a,b), considerably improve upon these models in sev-eral important respects, and in particular by carryingout calculations assuming spherical geometry. Moreover,the intensity calculations are based on a huge library ofatomic and molecular lines, with about 2 108 molecularlines contributing to a typical giant atmosphere model atTe = 3000 K. Claret (2000) also tted his new non-linearlimb darkening equation to some PHOENIX models.The dramatic dierence in the dependence of limbdarkening on emergent angle between the ATLAS mod-els and PHOENIX models is illustrated in Fig. 1, whichshows the intensity proles for a giant star of Te = 4250 Kand log g = 0:5, in four Johnson bands: V , R, I and K.The solid curve shows the PHOENIX proles, while thedashed, dash-dotted and dotted curves denote the linear,logarithmic and square root models respectively. It is clearthat there is a sharp decrease in the predicted intensityof the PHOENIX models close to the limb of the star,i.e. at  ’ 0:2. This feature does not appear in ATLASmodels and arises from the eects of spherical symmetry,which includes nite optical depth for rays close to thelimb { a treatment which is absent from ATLAS mod-els which utilise plane parallel atmospheres in which theFig. 1. Intensity proles for a giant star of Te = 4250 K andlog g = 0:5, in four Johnson passbands: V , R, I and K. Thesolid curve shows the PHOENIX proles, while the dashed,dash-dotted and dotted curves denote the ATLAS linear, log-arithmic and square root models respectively.optical depth of limb rays is still innite and so provide sig-nicant intensity out to  = 0. The question then arises: ismicrolensing suciently sensitive to detect this limb fea-ture in the atmospheres of extended sources, and thus totest the NextGen models against real observations?3. Computing microlensing lightcurvesFor a point source the amplication due to a point masslens is A =(u2 + 2=(upu2 + 4, where u(t) is the im-pact parameter, the projected angular separation betweenlens and source, measured in units of E, the angularEinstein radius of the lens, dened asE =s4GMc2(Ds −Dl)DsDl(1)whereDl andDs are the lens-observer and source-observerseparations respectively and M the mass of the lens. If theimpact parameter is comparable to the angular radius ofthe source, the point source amplication function breaksdown and it becomes necessary to calculate the amplica-tion as an integral over the source star, given byA(t) =R RI(r; ) A(r; ; t) r dr dR RI(r; ) r dr d (2)Thus, the microlensing lightcurve contains unique infor-mation on the source surface brightness prole, I(r; ),which is a far more sensitive test of the model atmospherethan its integrated value, the emergent flux.In the case of binary lenses, the amplication func-tion takes a more complex form, although for an extendedsource it is again given by an integral over the source.In this Letter we are only concerned with so-called foldcaustic crossings, and so we adopt the \square-root" ap-proximation for the amplication inside the caustic struc-ture (Schneider et al. 1992) which is valid within a fewLettertotheEditorH. M. Bryce et al.: Gravitational microlensing as a test of model atmospheres L3Fig. 2. A comparison of the V , R, I and K lightcurves pro-duced by the transit of a point lens with minimum impact pa-rameter u0 = 0:0 of a 4250 K, log g = 0:5, 0:1 E source, mod-elled with PHOENIX and linear limb darkening. Magnitudedierences are computed according to Eq. (3) in the text.Fig. 3. A comparison of the V , R, I and K lightcurves as inFig. 2, but now for an event with u0 = 0:09.angular source radii of the caustic and takes the formA(x) = A0 + 1=px. Here A0 is the total amplicationof other images outside the caustic, which we assume tobe constant during the caustic crossing, x is the perpen-dicular distance in units of the angular Einstein radiusof the binary lens, from an element of the source to thecaustic. Note that for elements of the source outside thecaustic no additional images are formed, so A(x) = A0.4. ResultsFigure 2 shows a comparison between two sets of mi-crolensing lightcurves for a point lens transit event: oneset calculated using a linear limb darkening law and pa-rameters from the ATLAS grid by Kurucz (1994) (seeValls-Gabaud 1998); and the second set calculated usingthe PHOENIX Next Generation models as discussed inSect. 2. In this example we compare the models for asource with of angular radius 0:1E, log g = 0:5, Te =4250 K and solar metallicity. The abscissa shows the timeFig. 4. A comparison of the V , R, I and K lightcurves pro-duced by fold caustic crossings of 4250 K, log g = 0:5, 0:1 Esources with PHOENIX and linear limb darkening accordingto Eq. (3).evolution of the event, in units of the Einstein crossingtime, tE, of the lens { i.e. the time taken for the lens tomove 1 angular Einstein radius. t0 denotes the time atwhich the impact parameter is a minimum, u0; in this ex-ample we take u0 = 0 { i.e. the lens transits through thecentre of the source. The ordinate shows the magnitudedierence between these models, dened asmagnextgen−maglinear =2:5 logFLLFUL−2:5 logFLNFUN(3)where FLL, FUL, FLN and FUN are the lensed and unlensedlinear limb darkened, and lensed and unlensed NextGenfluxes respectively. Four colour bands are used to illustratethe strong chromatic dierences between the models: V ,R, I and K bands, represented by straight, dashed, dash-dotted and dotted lines respectively.Two main features are immediately apparent inFig. 2. Firstly, just before and after the lens transitsthe source there are positive \spikes"; these occur asthe PHOENIX/NextGen model is considerably more limbdarkened than the linear model close to the limb. Howeverthese spikes are very narrow. The second feature is thebroad \brightening" during the central phase of the tran-sit; this also occurs as a consequence of the strongNextGen limb darkening, which makes the source appearsmaller as the lens crosses its centre, causing a largeramplication. Thus, if a source with a NextGen type at-mosphere is tted by a linear limb darkening model, thesource size would be systematically underestimated.Figure 3 compares lightcurves for the same source asFig. 2 but with an impact parameter of 0:09 E, i.e. thelens just transits the source. In this case the only fea-tures are the two upward spikes, due to the considerablysmaller NextGen intensity at the limb of the star. Again,however, these spikes are very narrow and around mini-mum impact parameter the dierence between the modelsis much smaller, since the amplied flux is dominated bya region slightly closer to the centre of the source wherethere is less dierence between the models (see Fig. 1 at = 0:9). In particular, no brightening eect is seen sincethe lens does not cross the central region of the source.LettertotheEditorL4 H. M. Bryce et al.: Gravitational microlensing as a test of model atmospheresThe amplitude of the eect is, however, similar: atleast 0:01 mag, which is readily detectable.Figure 4 compares the NextGen and linear models dur-ing a fold caustic crossing event, shown from when thesource is 3 source radii outside to 3 source radii insidethe caustic structure. Here the source has Te = 4250 K,log g = 0:5 and solar metallicity, but now has an an-gular radius of 0:01 E { a more realistic value for e.g.typical Bulge events, which results in larger amplicationchanges. The most striking feature in Fig. 4 is the largespike as the source begins to enter the caustic; here theflux is dominated by the small region of photosphere veryclose to the limb and directly beneath the caustic { thesurface brightness of which varies dramatically betweenthe two models. As the event proceeds the magnitude dif-ference becomes much smaller, particularly in mid-transitwhen the limb dierences between the models are dilutedby the larger intensity from the central part of the source.However, we again see a broad \brightening" eect as inFig. 2, due to the extreme NextGen limb darkening pro-ducing an apparently smaller source. As the trailing limbcrosses the caustic we see only a small peak since thereis already considerable amplication from the rest of thephotosphere. In practice, of course, we would expect to ob-serve these events in reverse: it is easier to predict whena source will exit a caustic than when it will enter!5. ConclusionsFigures 2{4 clearly suggest that it is within the capabili-ties of current intensive microlensing monitoring programsto discriminate between the chosen atmosphere modelsfrom a well parameterised lightcurve. For a point caus-tic event, with e.g. timescale tE ’ 15 days, the transitsshown in Figs. 2 and 3 would last for about 3 days, withthe sharp features at each limb crossing a few hours in du-ration. The sampling rates regularly achieved by e.g. thePLANET collaboration during recent events would easilysuce to resolve these features with a reasonable num-ber of data points. It must be noted, of course, that sinceFigs. 2 and 3 are for point lens transits, the probability ofsuch an event { and indeed of its immediate detection { issmall. Given that the event is alerted during the rise phaseas a potential extended source, however, the detection ofthe limb crossing spikes would be highly likely.The situation for the fold caustic event is more en-couraging. Again, the timescale for the caustic crossing isof the order of several hours, which is typical of \lensinganomaly" events already regularly monitored. Moreover,as pointed out previously, the chief advantage of fold caus-tic crossing events is the predictability of the source ex-iting the caustic structure, thus allowing intensive mon-itoring to be scheduled in advance. As the trailing limbof the star leaves the interior of the caustic one will see adramatic dierence between NextGen and the other mod-els since it is eectively only the (strongly darkened) limbof the star which is being amplied. The signature of thislimb crossing is seen in Fig. 4 to be of order 0.2 mag inJohnson B, V , R and I { which would render it very easilydetectable with existing photometric precision.To compare stellar atmosphere models using real datawould involve simultaneously determining best-t lens pa-rameters for the event. As remarked previously, Albrowet al. (2001) have included the eects of lens parametererrors in estimating limb darkening coecients. We willcarry out a similar numerical study, using realistic simu-lated observations, in future work and at a more advancedstage one must also consider the possibility of spots fur-ther eecting the lightcurve (see Hendry et al. 2002). Itseems clear, however, that the magnitude of the dierencebetween NextGen and ATLAS models is eminently de-tectable { even allowing for uncertainties in the tted lensparameters. This is an essential step in the determinationof fundamental stellar parameters. For instance, high ac-curacy measures of stellar radii via eclipsing binary lightcurves rely critically on these model atmospheres (e.g.Orosz & Hauschildt 2000) as do interferometric measures(e.g. Davis et al. 2000; Wittkowski et al. 2001). Thanksto microlensing we have now almost direct access to thedistribution of intensity across a source, and hence we canprobe stellar photospheres with unprecedented detail.ReferencesAlbrow, M. D., Beaulieu, J. P., Caldwell, J. A. R., et al. 1999,ApJ, 522, 1011Albrow, M. D., An, A., Beaulieu, J. P., et al. 2001, ApJ, 549,759Bogdanov, M. V., & Cherepashchuk, M. 1995, Ast. Lett., 21,505Bontz, R. J. 1979 ApJ, 233, 402Claret, A. 2000, A&A, 363, 1081Davis, J., Tango, W., & Booth, A. J. 2000, MNRAS, 318, 387Gould, A. 1994, ApJ, 421, L71Gould, A. 2001, PASP, 113, 903Hauschildt, P. H., Allard, F., & Baron, E. 1999a, ApJ, 512,377Hauschildt, P. H., Allard, F., Ferguson, J., Baron, E., &Alexander, D. R. 1999b, ApJ, 525, 871Hendry, M. A., Bryce, H. M., & Valls{Gabaud, D. 2002,MNRAS, in pressHeyrovsky, D., Sasselov, H., & Loeb, A. 2000, ApJ, 543, 406Kurucz, R. L. 1994, Atlas CDROMsemail: kurucz@cfa.harvard.eduNemiro, R. J., & Wickramasinghe, W. 1994, ApJ, 424, 21Orosz, J. A., & Hauschildt, P. H. 2000, A&A, 364, 265Rhie, S. H., & Bennett, D. P. 2002, ApJ, in press[astro-ph/9912050]Schneider, P., Ehlers, J., & Falco, E. E., 1992, GravitationalLenses (Springer-Verlag Berlin)Simmons, J. F. L., Willis, J. P., & Newsam, A. M. 1995, A&A,293, L46Valls-Gabaud, D. 1994, in Large Scale Structures in theUniverse, ed. J. Mu¨cket, S. Gottlober, & V. Muller(Singapore: World Scientic), 326Valls-Gabaud, D. 1998, MNRAS, 294, 747Van Hamme, W. 1993, AJ, 106, 2096Witt, H. J., & Mao, S. 1994, ApJ, 430, 505Witt, H. J. 1995, ApJ, 449, 42Wittkowski, M., Hummel, C. A., Johnston, K. J., et al. 2001,A&A, 377, 981LettertotheEditor

Atlas CDROMs email: kurucz@cfa.harvard.edu Nemiro,

Enlighten: Publications

We present calculations illustrating the potential of gravitational 
microlensing
to discriminate between classical models of 
stellar surface 
brightness profiles and the recently computed “Next Generation” models of
Hauschildt et al. These spherically-symmetric models include a much
improved treatment of molecular lines in the outer atmospheres of cool
giants – stars which are very 
typical sources
in Galactic bulge microlensing events. We show that the microlensing 
signatures of
intensively monitored point and fold caustic crossing events are readily 
able to distinguish
between NextGen and the classical models, provided a photometric
accuracy of 0.01 mag is reached. This accuracy is now routinely
achieved by alert networks, and hence current observations can
discriminate between such model atmospheres, providing a unique
insight on stellar photospheres

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Gravitational microlensing as a test of stellar model atmospheres

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