Although microlensing incorporates a fairly large number of parameters, most events can be understood quite intuitively. This glossary is intended as a quick reference, particularly to disambiguate the different symbol sets used by different authors over time. Interested readers are referred to the references at the bottom for a full discussion, especially Skowron et al. (2011), and to the Learning Resources menu.

NameCommonly-used symbolsUnitDefinition

Single Lens Parameters

Einstein crossing timetEdaysTime taken for the background source to cross the lens' Einstein radius, as seen by the observer. Caution: some early microlensing papers may refer to tE as the crossing time for the lens' Einstein diameter.
Time of peakt0daysTime at which the separation of lens and source reaches the minimum.
Source self-crossing timet*daysTime taken to cross the source's angular radius
$$t_{*} \equiv \rho t_{E}$$
Impact parameteru, at minimum u0DimensionlessThe angular separation, normalized to θE, between source and lens as seen by the observer
Conventionally u0 is positive when the lens passes to the right of the source star (Gould et al. 2004)
Effective timescaleteffdaysEqual to u0tE
RhoρDimensionlessThe angular source size θS normalized by the angular Einstein radius θE
Vector Microlens Parallax (also: annual parallax)π or π̄, components (πE,E, πE,N) or (πE,||, πE,⊥)The parallax to a lensing event caused by the motion of the Earth in its orbit during the event. $$\bar\pi_{E} = (\pi_{E,\parallel},\pi_{E,\perp})$$
E,NE,E)$$\bar\pi_{E} \equiv (\pi_{E,N},\pi_{E,E}) \equiv (\cos \phi_{\pi},\sin \phi_{\pi})\pi_{E},\, \rm{where}\, \pi_{E} = AU/\tilde{r_{E}}$$
E,||, πE,⊥)Components of parallax parallel and perpendicular to the apparent acceleration of the Sun, projected on the sky in a right-handed convention (Gould et al. 2004)$$\bar\pi_{E} = (\pi_{E,\parallel},\pi_{E,\perp})$$
Direction of lens motionφπradiansThe direction of lens motion relative to the source expressed as a counter-clockwise angle, north through east
Relative parallaxπrelRelative parallax observed for lens and source
$$\pi_{rel} = \theta_{E}\pi_{E}$$
Source parallaxπSParallax of the source star as seen from Earth
Lens distanceDLpcPhysical distance from the observer to the lensing object.
Source distanceDSpcPhysical distance from the observer to the source star.
Lens-source distanceDLSpcPhysical distance between the source and lens along the observer's line of sight.
Lens massMLMMass of the lensing object, including all component masses unless otherwise stated.
KappaκCommonly used to abbreviate equations for the mass of the lens, kappa gathers together all the physical constants in the equation:
$$\kappa = \frac{4G}{c^{2}AU}$$
Einstein angular radiusθEmasThe angle subtended by the Einstein radius of a lens from the distance of the observer.
Source angular radiusθ* or θSmasThe angle subtended by the source star radius at the distance of the observer
Einstein radiusREKmThe characteristic radius around the lens at which the images of the source form due to the gravitational deflection of light.
Projected Einstein radiusřEKmThe Einstein radius projected to the observer's plane.
Source radiusR* or RSKmThe physical radius of the source star
Helio- and geocentric proper motionsμhelio and μgeomas/yrProper motion of the source star relative to the Sun and Earth, respectively
$$\bar\mu_{geo} = \mu\bar\pi_{E}/\pi_{E}$$

Binary Lens Parameters

Parameter reference timet0,pardaysThe reference instant at which all parameters are measured in a geocentric frame that is at rest relative to the Earth at that time (An et al. 2002)
Fiducial timet0,kepdaysFiducial time specified during analysis of binary lens events. In general t0,kep and t0,par are defined to be equivalent
Lens massesM1,2,P or SM unless otherwise stated Most generically, the massive components of the lensing system are refered to as "M1" or "MP" for the primary or largest mass object and "M2" or "MS" for the secondary. In cases of a planet-star binary however, MP is sometime used to refer to the planet (i.e. secondary) while MS may refer to the star (primary)
Mass ratioqThe ratio of the masses of a binary lens, $$M_{2}/M_{1}$$
Mass fractionεThe ratio of the one of the masses in a binary lens to the total mass of that lens, $$M_{i}/M_{tot}$$
Lens separations, also s0, d or bDimensionlessThe projected separation of the masses of a binary lens during the event, normalized by the angular Einstein radius θE
Projected lens separationaAUProjected separation of binary lens masses in physical units.
Angle of lens motionα also α0radiansAngle (counter-clockwise) between the trajectory of the source and the axis of a binary lens, which is oriented pointing from the primary towards the secondary
Rate of change of lens separationds/dtθE/dayThe change in the projected separation of a binary lens due to the motion of the lens components in their orbit during an event
Rate of change of trajectory angledα/dtradians/dayThe change in the trajectory of the source relative to the axis of a binary lens, due to the orbital motion of the lens components during an event.
Earth orbital velocityv⊕,⊥km/sThe component of Earth's velocity at t0,par projected onto the plane of the sky
Binary lens orbital velocityγ or γ̄, components
Components of the velocity of the secondary lens relative to the primary due to orbital motion at time t0,kep
$$\gamma_{\parallel} = (ds/dt)/s_{0}, \gamma_{\perp} = -d\alpha/dt$$
γz is measured only in rare cases where the full Keplerian orbit can be determined (see Skowron et al. 2011), but is oriented such that positive γz points towards the observer
Binary lens orbital positionComponents
Components of the position of the secondary lens relative to the primary due to orbital motion at time t0,kep
The "perpendicular" component is always zero because the coordinate system is orientated with one axis along the binary axis.
Projected orbital velocityΔvProjected physical orbital velocity of the secondary of a binary lens relative to the primary
$$\bar\Delta v = D_{L}\theta_{E}s\bar\gamma$$
Projected orbital positionΔrProjected physical orbital position of the secondary of a binary lens relative to the primary
$$\bar\Delta r = D_{L}\theta_{E}(s,0,s_{z})$$
Lens plane coordinates(ξ,η)Coordinate system in the plane of a binary lens, parallel and perpendicular to the binary axis respectively
Orbital energyE⊥,kin,E⊥,potThe projected kinetic and potential energy due to binary lens orbital motion (Batista et al. 2011)
$$\frac{E_{\perp,kin}}{E_{\perp,pot}} = \frac{\kappa M_{\odot} \pi_{E}(|\bar\gamma|yr)^{2}s^{3}}{8\pi^{2}\theta_{E}(\pi_{E}+\pi_{S}/\theta_{E})^{2}}$$

Photometric Parameters

MagnificationA, at peak Amax or A0The magnification of the source star flux caused by the gravitational lens.
Event fluxf(t,k)counts/sThe total flux measured during a lensing event as a function of time, t, is the combination of the flux from the source being lensed plus the flux from (unlensed) background stars. Since different instruments, k, have different pixel scales and hence different degrees of blending, these are characterized with separate parameters. Commonly defined as:
$$f(t,k) = A(t)f_{S}(k) + f_{b}(k)$$
Source fluxfScounts/sFlux received from the source (as opposed to fb)
Blend fluxfbcounts/sFlux from background sources blended with the source.
Blend ratiogRatio of blend flux to source flux
Baseline magnitudeIbase or I0magThe measured brightness of a source star when unlensed, which may be blended with other stars
Peak magnitudeIpeakmagMeasured brightness of the source star at the time of smallest separation between lens and source, i.e. greatest brightness
Source magnitudeISmagMeasured (and reddened) source star magnitude
Dereddened source magnitudeIS,0magSource star magnitude when corrected for interstellar reddening
Blend magnitudeIBmagMeasured magnitude of stars blended with the source star
Lens magnitudeIL,HLmagMagnitude of the lens star measured in I and H passbands
Source star colorUsually (V-I)SmagMeasured color (here in (V-I) bands) of the blended and reddened source star
Dereddened source colorUsually (V-I)S,0magDereddened color of the source star
Blend colorUsually (V-I)BmagThe combined color of stars blended with the source
Extinction coefficientUsually AImagExtinction between the observer and the source star, here in the I passband
Reddening cofficientUsually E(V-I)magReddening term between the observer and the source, here in the V and I passbands
Limb darkening coefficientΓλmagLimb darkening coefficient for passband λ
Limb darkening coefficientuλmagLimb darkening coefficient for passband λ

Key Concepts

Optical depthτstar-1The probability that a given star, at a specific instant in time, has an amplification caused by gravitational microlensing of A ≻ 1.34. This is the fraction of a given solid angle of sky observed which is covered by the Einstein rings of all lensing objects within that area.

Naming Convention

Microlensing events are usually named for the survey which independently discovered them, the year in which they were discovered and the region of the sky in which they were found.
For example: OGLE-2017-BLG-1234 refers to the 1234th event found by the OGLE survey in the Galactic Bulge during the 2017 observing season.
As most microlensing events are found in largely the same region of the Galactic Bulge, it is often the case that multiple surveys will find the same event independently. In these cases, it is customary for the event to be refered to by a joint name, in the sequence in which public alerts were issued.
For example: OGLE-2017-BLG-1234/MOA-2017-BLG-234 indicates that OGLE issued a public alert first, and MOA subsequently found the same event independently.


An et al. (2002), MNRAS, 356, 1409
Skowron et al. (2011), ApJ, 738, 87
Gould, A. (2000), ApJ, 542, 785
Gould et al. (2004), ApJ, 614, 404
Batista et al. (2011), A&A, 529, 102