If the standard microlensing geometry is inverted so that the Einstein ring
is projected onto the observer plane rather than the source plane, then the
relations between the observables (\theta_E,\tilde r_E) and the underlying
physical quantities (M,\pi_rel) become immediately obvious. Here \theta_E and
\tilde r_E are the angular and projected Einstein radii, M is the mass of the
lens, and \pi_rel is the lens-source relative parallax. I recast the basic
formalism of microlensing in light of this more natural geometry and in terms
of observables. I then find that the relations between observable and physical
quantities assume an exceptionally simple form. In an appendix, I propose a set
of notational conventions for microlensing.Comment: 8 pages, 1 figure tells all. Interested parties are requested to vote
  on a proposed standard for microlensing notation given in the appendix.
  Submitted to Ap

Andrew Gould

Bennett D. B.

Mao S.

Refsdal S.

English

arXiv

If the standard microlensing geometry is inverted so that the Einstein ring is projected onto the observer plane rather than the source plane, then the relations between the observables (θ_E,r_E) and the underlying physical quantities (M,πrel) become immediately obvious. Here θ_E and r_E are the angular and projected Einstein radii, M is the mass of the lens, and π_rel is the lens-source relative parallax. I recast the basic formalism of microlensing in light of this more natural geometry and in terms of observables. I then find that the relations between observable and physical quantities assume an exceptionally simple form. In an appendix, I propose a set of notational conventions

Gould, Andrew

KnowledgeBank at OSU

THE ASTROPHYSICAL JOURNAL, 542 :785È788, 2000 October 202000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(A NATURAL FORMALISM FOR MICROLENSINGANDREW GOULDDepartment of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210 ; gould=astronomy.ohio-state.eduReceived 2000 January 24 ; accepted 2000 June 1ABSTRACTIf the standard microlensing geometry is inverted so that the Einstein ring is projected onto the obser-ver plane rather than the source plane, then the relations between the observables and the under-(hE, r8 E)lying physical quantities become immediately obvious. Here and are the angular and(M, nrel) hE r8 Eprojected Einstein radii, M is the mass of the lens, and is the lens-source relative parallax. I recastnrelthe basic formalism of microlensing in light of this more natural geometry and in terms of observables. Ithen Ðnd that the relations between observable and physical quantities assume an exceptionally simpleform. In an appendix, I propose a set of notational conventions.Subject headings : astrometry È gravitational lensing1. INTRODUCTIONThe geometry of point-lens microlensing (Einstein 1936 ;Refsdal 1964 ; 1986) is so simple that students canPaczyn skiderive all the basic results in a few hours. Nonetheless, thisgeometry has never been boiled down to its essence : therelationship between the underlying physical quantities andthe observables. In particular, the ““ Einstein ring radius ÏÏ rE,a central concept in the usual formulation, is not directlyobservable and has not been observationally determined foreven one of the D500 microlensing events observed to date.There appear to be three reasons that the natural geometricformulation has not been developed. First, the standardgeometry is already so trivial that further simpliÐcation hasnot seemed worthwhile. Second, the theory of microlensingwas already quite developed before it was realized what theobservables were, and until very recently the prospects werepoor for measuring these observables except in a handful ofevents. Third, the original impulse to microlensing searcheswas to probe the dark matter. This focused attention on theoptical depth (a statistical statement about the ensemble ofevents) and secondarily on the Einstein timescale which,tE,of the three observables, is the one that has the most convo-luted relation to the underlying physical parameters.However, with the prospect of astrometric microlensing itis now possible that a second observable, the angular Ein-stein radius will be routinely measured (Boden, Shao, &hE,Van Buren 1998 ; 1998). Moreover, if thesePaczyn skiastrometric measurements are carried out by the SpaceInterferometry Mission (SIM) in solar orbit, then compari-son of photometry from SIM and the ground will yield athird observable, the projected Einstein radius (Refsdalr8 E1966 ; Gould 1995 ; Gould & Salim 1999). Hence, it is nowappropriate to reformulate the microlensing problem interms of these observables.2. GEOMETRYThe upper panel of Figure 1 shows the standard presen-tation of microlensing geometry (e.g., Fig. 3 from Gould1996). The observer (O), lens (L) of mass M, and source (S)are aligned. The light is deÑected by an angle a given by theEinstein (1936) formulaa \ 4GMrE c2, (1)where is the Einstein radius. It arrives at the observerrEdisplaced by an angle from the true position of thehEsource. In this case, the source is therefore imaged into aring. The size of this ring projected onto the source plane isMore generally, the alignment will not be perfect, andrü E.the axial symmetry will be broken. Hence, there will be twoimages rather than a ring. However, even in this moregeneral case the Einstein ring provides a natural scale to theproblem.The lower panel of Figure 1 basically inverts thegeometry of the upper panel and thereby focuses attentionon the observer rather than the source. This seems like atrivial change, but it has two advantages. First, the quan-tities shown at the right, and are the observables. TohE r8 E,date, has been measured for only eight events (Alcock ethEal. 1997, 2000 ; Albrow et al. 1999, 2000a, 2000b ; Afonso etal. 2000), all by using the source as an ““ angular ruler ÏÏ(Gould 1994a ; Nemiro† & Wickramasinghe 1994 ; Witt &Mao 1994). Similarly, has been determined for onlyr8 Eabout a half-dozen events (Alcock et al. 1995 ; Bennett et al.1997 ; Mao 1999). For all of these, was found by measur-r8 Eing the deviation of the light curve induced by the EarthÏsmotion (Gould 1992). The amplitude of this deviation isproportional to The measurements of bothnE4 AU/r8 E. hEand have required special conditions (a caustic crossingr8 Efor and an event lasting a large fraction of a year forhE r8 E),which is why so few of these ““ observables ÏÏ have actuallybeen observed. However, as mentioned above, both andhEcould be measured routinely in the future.r8 E The second reason for inverting the standard geometry isthat doing so makes transparent the relation between theobservables and the underlying physical variables : theproduct of and is essentially the Schwarzschild radiushE r8 Eof the lens, and their ratio is essentially the lens-source rela-tive parallax. Using the small-angle approximation, onesees immediately from the lower panel of Figure 1 thatora/r8 E\ hE/rE,hE r8 E\ arE\4GMc2 . (2)Next, from the exterior-angle theorem,hE\ a [ t\r8 EDl[ r8 EDs\ r8 EDrel, (3)785786 GOULD Vol. 542FIG. 1.ÈAbove : Standard microlensing geometry. The bold curveshows the path of the light from the source (S) to the observer (O) beingdeÑected by the lens (L) of mass M. The deÑection angle is a \ 4GM/rE c2,where is the Einstein radius shown as a dashed line. The image (I) isrEdisplaced from the source by the angular Einstein radius which, pro-hE,jected onto the source plane, corresponds to a physical distance Below :rü E.Natural microlensing geometry. Mostly the same as the upper panel,except that the Einstein radius is now projected onto the observer plane asrather than onto the source plane as This minor di†erence allows oner8 E rü E.to see immediately the relations between the observables and the(hE, r8 E)physical parameters (M, First, under the small-angle approximation,nrel).so Second, by the exterior-anglea/r8 E\ hE/rE, r8 E hE\ arE\ 4GM/c2.theorem, where and are the distances tohE\ a [ t\ r8 E/Dl[ r8 E/Ds, Dl Dsthe lens and source. Hence, where is the lens-sourcehE/r8 E\nrel/AU, nrelrelative parallax.where and are the distances to the lens and source,DlDsand Note that equation (3) can beDrel~1 4Dl~1[Ds~1.written more suggestively asnE hE\ nrel , nE4AUr8 E, (4)where is the lens-source relative parallax.nrel\ AU/DrelJust as in astrometric parallax determinations, where n isa more natural way to represent the measured quantitythan its inverse (distance), so in microlensing ““ parallax ÏÏdeterminations is more natural than its inverse ThenE (r8 E).reason is the same : the observable e†ect is inversely pro-portional to but directly proportional to so the mea-r8 E nE,surement errors when expressed in terms of exhibit morenEregular behavior. As in the case of astrometric parallax, thisfeature becomes especially important for measurementsthat are consistent with zero at the few-p level. Indeed, incontrast to astrometric parallaxes, microlensing parallaxesare inherently two-dimensional (Gould 1995). That is, onemeasures not only the amplitude of (or but also ther8 E nE)direction of lens-source relative motion. Hence one can gen-eralize to a two-dimensional vector whose direction isnE pEthat of the lens relative to the source. The measurementerrors in are then easily expressed as a covariancepEmatrix. By contrast, there is no natural way to generalize r8 E :it can be made into a vector with the same direction butr8 E,when is consistent with zero, such a vector is very poorlypEbehaved. Moreover, in some cases one component of canpEbe very well determined while the other is highly degenerate(Refsdal 1966 ; Gould 1994b, 1995), a situation that is easilyrepresented using but unwieldy using (Note thatpE r8 E.while no one has ever previously introduced the vector Ir8 E,have often discussed the closely related projected velocityvector, ¿8 \ r8 E/tE.)The Einstein crossing time is the only observable thattEat present is routinely observed. While I Ðnd no fault withconsiderations of symmetry with the substitutiontE, lead me to substitute wherer8 E] pE tE] lE,kE41tE, (5)and where the direction of is that of the lens motionlErelative to the source. With this deÐnition, the relative lens-source proper motion is given by lrel \ lE hE.3. RELATIONS BETWEEN OBSERVABLES AND PHYSICALQUANTITIESFrom equations (2)È(4), one immediately derivesr8 E\S4GMDrelc2 , nE\SnreliM(6)andhE\S 4GMDrel c2\ JiMnrel ,(7)wherei 44Gc2AU\4v2^M_c2 ^ 8.144masM_, (8)and km s~1 is the speed of the Earth.v^D 30How well is the coefficient (8.14 . . . ) in i known? Itsu†ers from two sources of uncertainty. First, the factor 4 inequations (8) and (1) is a prediction of general relativity(GR). Its accuracy (often parameterized by c) has been veri-Ðed experimentally by Hipparcos, but only to 0.3%(Froeschle, Mignard, & Arenou 1997). However, if GR isassumed to be exact, then this coefficient can be determinedas accurately as which should be known from pulsar(v^/c)2,No. 2, 2000 NATURAL FORMALISM FOR MICROLENSING 787timing and solar system radar ranging to at least nine sig-niÐcant digits.In astrometric microlensing measurements, one auto-matically recovers the parallax and proper motion of thesource, and (Boden et al. 1998 ; Gould & Salim 1999).nslsHence, the observables are and WhenlE, pE, hE, ns ls.expressed in this natural form, they have a particularlysimple relation to the physical properties of the lens :M \ hEinE, (9)nl\ nE hE] ns , (10)ll\ lE hE] lE , (11)and¿M,l\lE hE] lsnE hE] ns, (12)where and are the parallax, proper motion, andnl, ll, ¿M,ltransverse velocity of the lens.I thank A. Babul, D. Bennett, N. Dalal, K. Griest, E.Kerins, and S. Rhie for useful comments and clariÐcations.This work was supported by grant AST 97-27520 from theNSF.APPENDIXTHE NEED FOR UNIFORM NOTATIONMicrolensing su†ers from a plethora of mutually inconsistent notational conventions. While this poses no real problem forveterans, it presents signiÐcant obstacles to newcomers entering the Ðeld. I take the opportunity of this paper (which, morethan most, concerns itself with notational issues) to try to forge a consensus. In formulating my proposed conventions, I aminÑuenced primarily by prevalence of current usage and secondarily by the need for internal consistency.I am abandoning some of my own prized notations, and I hope others are willing to do the same in the interest of achievinga uniform system. A number of comments I received after a preprint of this paper appeared give grounds for cautiousoptimism.First, all quantities associated with the size of the Einstein ring (in units of length, angle, time, etc.) should be subscriptedwith an uppercase roman ““ E ÏÏ in conformity with ApJ conventions. All physical Einstein radii should be denoted r. Hence, r8 E,and for the Einstein rings in the planes of the observer, lens, and source. The other quantities are for the angularrE, rü E hEEinstein radius, for the Einstein radius crossing time, and the direction of and deÐned by thetE kE4 tE~1, nE4 AU/r8 E, pE lEdirection of the proper motion of the lens relative to the source. Note that the Einstein diameter crossing time is by commonconvention denoted tü .Second, all quantities associated with position in the Einstein ring should be denoted by u, or possibly by u if a vectorposition is indicated. When u is a vector, it must be speciÐed whether it is the source position relative to the lens or vice versa.Common usage seems to conform to the former, and hence I adopt that. However, keep in mind that this means thatdu/dt \ [lrel/hE.Third, all quantities associated with the time of closest approach to the center of the Einstein ring should be denoted by asubscript zero : thus, for the time of closest approach and for the projected separation of the lens and source in units oft0 u0 hEat time t0.Fourth, all quantities associated with the source should be denoted by a subscript asterisk : thus, for the angular radiush*of the source and for its physical radius.r*Fifth, time normalized to the Einstein crossing time should be denoted Hence the vector position in theqE\ (t[ t0)/tE.Einstein ring is u \ (qE, u0).Sixth, the distances to the lens and source should be denoted andDlDs.Seventh, event parameters as measured from locations other than the Earth should be subscripted, e.g., for the time““ t0,s ÏÏof closest approach as seen from a satellite. The subscript Sun symbol (x) should be reserved for event parameters as seenfrom the Sun (not in the Sun frame but from another location). The subscript Earth symbol (=) should be used only whenneeded to avoid confusion.Finally, the reader will note that I have described di†erent parameters that contain the same information, e.g., and(r8 E, nE)I expect that and will come into use mainly in technical applications, and that the general reader of microlensing(tE, kE). pE lEarticles will continue to Ðnd and to be more intuitive. In particular, in cases where there is only a microlensing parallaxr8 E tEmeasurement, the projected velocity is often a substantially more useful representation of the measurement than¿8 \ lE/nE lEand reported separately. Note that in contrast to which represents two components of an intrinsicallynE ¿l,M\ ll/nl,three-dimensional vector, is intrinsically two-dimensional and so should not be subscripted with a ““ perpendicular ÏÏ symbol¿8(o).REFERENCESAfonso, C., et al. 2000, ApJ, 532, 340Albrow, M., et al. 1999, ApJ, 522, 1011ÈÈÈ. 2000a, ApJ, 534, 894ÈÈÈ. 2000b, ApJ, submitted (astro-ph/0004243)Alcock, C., et al. 1995, ApJ, 454, L125ÈÈÈ. 1997, ApJ, 491, 436Alcock, C., et al. 2000, ApJ, 541, 270Bennett, D. B., et al. 1997, BAAS, 191.8303Boden, A. F., Shao, M., & Van Buren, D. 1998, ApJ, 502, 538Einstein, A. 1936, Science, 84, 506Froeschle, M., Mignard, F., & Arenou, F. 1997, Proc. ESA Symp.,HipparcosÈVenice Ï97 (ESA SP-402 ; Paris : ESA), 49788 GOULDGould, A. 1992, ApJ, 392, 442ÈÈÈ. 1994a, ApJ, 421, L71ÈÈÈ. 1994b, ApJ, 421, L75ÈÈÈ. 1995, ApJ, 441, L21ÈÈÈ. 1996, PASP, 108, 465Gould, A., & Salim, S. 1999, ApJ, 524, 794Mao, S. 1999, A&A, 350, L19Nemiro†, R. J., & Wickramasinghe, W. A. D. 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