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For a single lens: we got

\begin{equation} u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2 + u_0^2},\end{equation}

so the magnification is

\begin{equation} A = \frac{u^2+2}{u\sqrt{u^2+4}} \end{equation}

Therefore the magnification for a single lens depends on three parameters: $u_0$ (minimum separation lens-source), $t_0$ (time where the magnification is maximum) and $t_E$ is the characteristic event timescale, which is the time it takes for the source to move with respect to the lens by one Einstein ring radius.

For a binary lens, however, the formulae of $u$ and $A$ above no longer hold. $A$ can be calculated for a point $u=(u_1,u_2)$ using numerical methods (Press et al. 2002) but, how is calculated $u$ for a certain time $t$? According to Tsapras (2018) $u$ depends on $u_0$, $t_E$, $t_0$ but also the angle $\alpha$ at which the source trajectory crosses the binary axis, the mass ratio between the two components, $q$, and the separation between them, $s$, projected on the lens plane. ¿How these last three parameters come into play in the calculation of $u$ for a binary lens?

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This answer applies to the simplest case: a static binary and a point source.

In the binary case, $u(t)$ describes how close each point in the source trajectory comes to (what you have defined as) the origin of the binary axis.

The solution is non-trivial. It involves finding the roots of a 5th order complex polynomial. You can look for example at how Etienne Bachelet's pyLIMA package codes it in Python. If you want to do it yourself, you first need to identify the location of the caustics and critical curves for the given arrangement. For that you need your two masses (or just the mass ratio) and their separation in units of the Einstein radius. The coefficients of the polynomial (A, B, C) are described in Schneider & Weiss 1986 (eqn 9b).

Then you need to define the trajectory of your source relative to the binary axis. You also have to choose your origin (some people use the midpoint between the masses, others use the largest of the two masses or the barycenter).

For each point in the trajectory (instantaneous source position) you have to evaluate the corresponding magnification, that is, find the roots of the polynomial for that position. You will then get five solutions, which correspond to the images of the source. However, not all 5 solutions correspond to real images. To find out which are the real ones, you have to substitute them in the inverted equation and see if they map back to the given source position. You will always have three (outside the caustics) or five (inside the caustics) real images.

You can see an example below. Equal mass binary

You can see how different source trajectories generate different light curves at this link.

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The projected separation of the lens and source in the lens plane ($u(t)$), is described as you give it. In the case of binary lenses, since they have two masses separated by some distance $s$ in the lens plane, $u(t)$ is usually broken down and defined relative to the axis of the binary (the line connecting the two components). For convenience, the calculations are usually made using coordinates of $\tau(t)$ (the component of $u(t)$ along the binary axis) and $\beta(t)$ (the component perpendicular to the binary axis), with the source trajectory intersecting this binary axis at the angle $\alpha$, as you describe.

However, the masses in binary lenses are themselves in orbit about one another, and so we can't always consider this to be a static reference frame. Those orbits naturally depend on the lens masses and their Keplerian motion around the center of mass of the binary. This is taken into account by recognizing that $s$ and $\alpha$ are themselves functions of time: $$ s = s_0 + ds/dt\\ \alpha = \alpha_0 + d\alpha/dt $$ ...and these additional parameters ($ds/dt$ and $d\alpha/dt$) are fitted during the lightcurve modeling process, along with the rest of the microlensing parameters. This is covered in more detail in Batista et al. (2011) (Section 4) which gives a clear description of these calculations.

It is also worth pointing out that binary orbital motion is not the only factor which modifies the calculation of $u(t)$ - parallax is very important as well, and can lead to degeneracies in the modeling. The most common cause is the motion of the Earth during the microlensing event. This is a substantial topic, but for an overview, and a list of relevant citations, see the parallax section of the microlensing tutorial.

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  • $\begingroup$ That doesn't respond my answer. First of all, I assume that $s$ and $\alpha$ are constant, so I don't mind their derivatives. Second, I don't take account of the parallax effect. What I want is a formula to calculate $u(t)$, for a given $u_0$, $\alpha$,$t_0$ and $t_E$. $\endgroup$ – Carlos Vázquez Monzón Mar 7 at 20:30

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