Microlensing has quite a lot of parameters! This glossary is intended to disambiguate the different symbol sets used by different authors over time. For a full discussion of microlensing parameters, see Skowron et al. (2011), ApJ, 738, 87.

 

Single Lens Parameters

Name Commonly-used symbols Unit Definition
Einstein crossing time tE days Time 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</i>
Time of peak t0 days Time at which the separation of lens and source reaches the minimum.
Source self-crossing time t* days Time taken to cross the source’s angular radius
Impact parameter u, at minimum u0 Dimensionless The 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 timescale teff days Equal to u0tE
Rho ρ Dimensionless The 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.
Direction of lens motion φπ radians The direction of lens motion relative to the source expressed as a counter-clockwise angle, north through east
Relative parallax πrel   Relative parallax observed for lens and source
Source parallax πS   Parallax of the source star as seen from Earth
Lens distance DL pc Physical distance from the observer to the lensing object.
Source distance DS pc Physical distance from the observer to the source star.
Lens-source distance DLS pc Physical distance between the source and lens along the observer’s line of sight.
Lens mass ML M⊙ Mass of the lensing object, including all component masses unless otherwise stated.</td>
Kappa κ   Commonly used to abbreviate equations for the mass of the lens, kappa gathers together all the physical constants
Einstein angular radius θE mas The angle subtended by the Einstein radius of a lens from the distance of the observer.
Source angular radius θ* or θS mas The angle subtended by the source star radius at the distance of the observer
Einstein radius RE Km The characteristic radius around the lens at which the images of the source form due to the gravitational deflection of light.
Projected Einstein radius řE Km The Einstein radius projected to the observer’s plane.
Source radius R* or RS Km The physical radius of the source star
Helio- and geocentric proper motions μhelio and μgeo mas/yr Proper motion of the source star relative to the Sun and Earth, respectively

Binary Lens Parameters

Name Commonly-used symbols Unit Definition
Parameter reference time t0,par days The 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 time t0,kep days Fiducial time specified during analysis of binary lens events. In general t0,kep and t0,par are defined to be equivalent
Lens masses M1,2,P or S M⊙ unless otherwise stated  
Mass ratio q   The ratio of the masses of a binary lens, M2/M1
Mass fraction ε   The ratio of the one of the masses in a binary lens to the total mass of that lens, Mi/Mtot
Lens separation s, also s0, d or b Dimensionless The projected separation of the masses of a binary lens during the event, normalized by the angular Einstein radius θE
Projected lens separation a⊥ AU Projected separation of binary lens masses in physical units.
Angle of lens motion α also α0 radians Angle (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 separation ds/dt θE/year The 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 angle dα/dt radians/year The 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 velocity v⊕,⊥ km/s The component of Earth’s velocity at t0,par projected onto the plane of the sky
Binary lens orbital velocity γ or γ̄, components
(γ∥,γ⊥,γz)		   Components of the velocity of the secondary lens relative to the primary due to orbital motion at time t0,kep
Binary lens orbital position Components
(s,0,sz)		   Components of the position of the secondary lens relative to the primary due to orbital motion at time t0,kep
Projected orbital velocity Δv   Projected physical orbital velocity of the secondary of a binary lens relative to the primary
Projected orbital position Δr   Projected physical orbital position of the secondary of a binary lens relative to the primary
Lens plane coordinates (ξ,η) Normalized to θE Coordinate system in the plane of a binary lens, parallel and perpendicular to the binary axis respectively

Photometric Parameters

Name Commonly-used symbols Unit Definition
Magnification A, at peak Amax or A0   The magnification of the source star flux caused by the gravitational lens.
Event flux f(t,k) counts/s The total flux measured during a lensing event as a function of time, t, from instrument, k
Source flux fS counts/s Flux received from the source (as opposed to fb)
Blend flux fb counts/s Flux from background sources blended with the source.
Blend ratio g   Ratio of blend flux to source flux
Baseline magnitude Ibase or I0 mag The measured brightness of a source star when unlensed, which may be blended with other stars
Peak magnitude Ipeak mag Measured brightness of the source star at the time of smallest separation between lens and source, i.e. greatest brightness
Source magnitude IS mag Measured (and reddened) source star magnitude
Dereddened source magnitude IS,0 mag Source star magnitude when corrected for interstellar reddening
Blend magnitude IB mag Measured magnitude of stars blended with the source star
Lens magnitude IL,HL mag Magnitude of the lens star measured in I and H passbands
Source star color Usually (V-I)S mag Measured color (here in (V-I) bands) of the blended and reddened source star
Dereddened source color Usually (V-I)S,0 mag Dereddened color of the source star
Blend color Usually (V-I)B mag The combined color of stars blended with the source
Extinction coefficient Usually AI mag Extinction between the observer and the source star, here in the I passband
Reddening cofficient Usually E(V-I) mag Reddening term between the observer and the source, here in the V and I passbands
Limb darkening coefficient Γλ mag Limb darkening coefficient for passband λ (An et al. 2002)
Limb darkening coefficient uλ mag Limb darkening coefficient for passband λ

Key Concepts

Name Commonly-used symbols Unit Definition
Optical depth τ star-1 The probability that a given star, at a specific instant in time, has an magnification 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.
Event rate Γ star-1 yr-1 The rate at which microlensing occurs.