*by Rachel Street*

In the course of exploring point-source, point-lens models, we've made a number of useful but simplifying assumptions. For many events, these assumptions actually serve us well. In the real world of course, things are not quite so straight forward, and some events do deviate from PSPL without actually being binary or variable. Here we examine the most significant causes of deviations, and introduce some new parameters which will be relevant when we consider binary events.

Assumption No. 1: The Earth is a stationary observing platform through-out the event, and the relative movements of the lens and source can be considered to be plane parallel to the sky.

Of course, these things are not true. All three bodies are moving, including the Earth around the Sun, which can only be considered to be a straight line over small timescales. For some events, particularly long timescale events, this assumption breaks down, and the resulting parallax must be taken into account.

The effect of parallax is that the observer's "perspective" on the lens-source separation changes more quickly/slowly than expected, depending on the relative directions of motion. Since:

$$A = \frac{u^{2}+2}{u(u^{2}+4)^{1/2}}$$ | [24] |

depends on this separation, u = α/θ_{0}, the observer sees variations in the amplification from that
expected from a purely PSPL event.

Eqn. 24 depends only on the impact parameter - the apparent separation of lens and source at a given time. The characteristic
parameters of the event - M_{L}, D_{L}, D_{S} etc - are degenerate since together they define θ_{0}
and many combinations of these parameters can produce the same amplification. Therefore, parallax can be useful because the
variations it causes give us additional information regarding the relative movements of the bodies, which can be used to break
degeneracies in the model, and learn more about the characteristics of the system.

The Einstein crossing time t_{E} (time taken to cross the Einstein radius) is given as:

$$t_{E} = \frac{R_{E}}{\bar{\nu}} = \frac{D_{L}\theta_{0}}{\bar{\nu}} = \omega^{-1}$$ |

where ν̄ is the vector of relative motion and ω^{-1} is referred to as the vector event rate. So far, we've
consider the separation at time t, u_{☉}(t), neglecting Earth's orbital motion, which can be represented as:

$$u_{\odot}(t) = \beta + \omega(t - t_{0})$$ | [26] |

where β = b/D_{L}θ_{0} is the normalized impact parameter for a minimum physical separation of the
lens from the Sun-Source line of sight, b. This minimum separation occurs at the peak amplification of the event, at time t_{0}.

However, for an observer on Earth, u_{⊕}(t), must take into account our annual orbital motion, which is known
relative to background stars from celestial mechanics:

$$u_{\oplus}(t) = u_{\odot}(t) + \pi_{E}\hat{\beta} \cos(\Omega(t - t_{0}) + \phi) + \Lambda \hat{\omega}(\sin(\Omega(t - t_{0}) + \phi)$$ | [27] |

where Ω = 2 π/yr is Earth's orbital motion, and φ is our phase in that motion at t = t_{0} relative to the
normalized impact parameter at Earth, β̂. Λ is the parity ≡ -j̄(β̂ × ω̂) = ∓1, where
j̄ is the unit vector toward the North ecliptic pole.

π_{E} is the microlensing parallax parameter = a_{⊕}/ř_{E}, where a_{⊕} is
1 AU and ř_{E} is the Einstein radius projected onto the observer plane (aka the reduced Einstein radius).

$$\tilde{r_{E}} = \theta_{0}D_{L} = \left( \frac{4GM_{L}D_{rel}}{c^{2}} \right)^{\frac{1}{2}}$$ | [28] |

$$D_{rel} = \frac{1}{1/D_{L} - 1/D_{S}}$$ |

Eqn. 27 can then be substituted into eqn. 24 to plot the parallax-modified amplification during the event.

Given the known sky position of the event and the Earth's position in it's orbit at the time, most of the additional parameters are actually calculable,
so in principle the parallax introduces one additional parameter - π_{E} into the fitting process.

However, in reality, π_{E} has 2 components - one which is parallel to the Sun at to and one perpendicular.
π_{E,||} is relatively easy to spot as it introduces asymmetries in the lightcurve around the peak.
π_{E,⊥} however introduces symmetric distortion and can be more difficult to distinguish observationally, as it
mimics a different set of event parameters.

Gould, A. (2000), ApJ, 542, 785

Buchalter, A. & Kamionkowski, M. (1997), ApJ, 482, 782