BinarySources.md

Back to Orbital motion

Binary sources

Binary sources just give the superposition of two single-source microlensing light curves. In VBMicrolensing we have the BinSourceLightCurve function, illustrated in the following example:

VBMicrolensing VBM; // Declare instance to VBMicrolensing

double pr[6]; // Array of parameters
double tE, FR, u01, u02, t01, t02, t;

u01 = 0.01; // Impact parameter for source 1
u02 = 0.03; // Impact parameter for source 2
t01 = 7550.4; // Time of closest approach to source 1
t02 = 7551.8; // Time of closest approach to source 2
tE = 37.3; // Einstein time
FR = 0.1; // Flux ratio of the second source to the first

// Let us put all parameters in our array
pr[0] = log(tE);
pr[1] = log(FR);
pr[2] = u01;
pr[3] = u02;
pr[4] = t01;
pr[5] = t02;

t = 7551.6; // Time at which we want to calculate the magnification

Mag = VBM.BinSourceLightCurve(pr, t); // Calculates the Binary Source magnification at time t with parameters in pr
printf("Binary Source Light Curve at time t: %lf", Mag); // Output should be 29.97...

The output of BinSourceLightCurve is a magnification compared to the baseline flux. Therefore, it is the sum of two Paczynsky light curves weighted by 1/(1+FR) and FR/(1+FR) respectively.

Parallax is included in the BinSourceLightCurveParallax function, which accepts two more parameters for the parallax components, as illustrated in the Parallax section.

Extended binary sources

If the finite size of the sources is relevant, one can use BinSourceLightCurve function

rho = 0.01; // Size of source 1
pr[6] = log(rho); 
Mag = VBM.BinSourceExtLightCurve(pr, t); // Calculates the magnification for extended binary sources

Only one source size is specified as an independent parameter, while the source size of the second source is obtained through mass-radius-luminosity relations. This ensures that the user has full control on the physical consistency of the model.

Mass-radius-luminosity relations for binary sources

The mass-luminosity relation in VBMicrolensing is a power law of the form $L \sim M^q$ where the exponent $q$ is given by the variable VBM.mass_luminosity_exponent, whose default value is $4.0$.

The mass-radius relation is a power law of the form $\rho \sim M^p$ where the exponent $p$ is given by the variable VBM.mass_radius_exponent, whose default value is $0.9$.

Therefore, in the function BinSourceExtLightCurve, if the flux ratio is FR and the radius of the first source is rho, the radius of the second source is calculated as rho * FR^{p/q}.

The user can customize the two exponents by changing VBM.mass_luminosity_exponent and VBM.mass_radius_exponent as appropriate for the sources in the specific microlensing event and for the observation band.

Xallarap

Binary sources can also orbit around a common center of mass. VBMicrolensing offers xallarap with circular orbital motion, described by 6 parameters:

$(\xi_\parallel, \xi_\perp)$, projections of the node lines parallel and perpendicular to the source velocity at time $t_0$. Note that the orbital radius in Einstein angle units is $\sqrt{\xi_\parallel^2 + \xi_\perp^2}$;

$\omega = \frac{2\pi}{T}$, orbital angular velocity;

$i$, inclination of the orbit;

$\phi_0$, phase at time $t_0$ from the line of nodes;

$q_s$, mass ratio of the two source components.

Here is an example with the function BinSourceSingleLensXallarap. You may note that the parametrization of the sources is very different with respect to the previous functions.

VBMicrolensing VBM; // Declare instance to VBMicrolensing
VBM.LoadESPLTable("ESPL.tbl");

double pr[10]; // Array of parameters
double u0, t0, tE, rho, xi1, xi2, om, inc, phi0, qs, t, Mag;

u0 = 0.01; // Impact parameter for the first source
t0 = 7550.4; // Time of closest approach for the first source
tE = 37.3; // Einstein time
rho = 0.004; // Radius of the first star
xi1 = 0.011; // Xallarap component 1
xi2 = 0.02; // Xallarap component 2
om = 0.04; // Orbital velocity
inc = 0.8; // Inclination
phi0 = 1.4; // Phase from the line of nodes
qs = 0.1; // Mass ratio of the two stars

// Let us put all parameters in our array
pr[0] = u0;
pr[1] = t0;
pr[2] = log(tE);
pr[3] = log(rho);
pr[4] = xi1;
pr[5] = xi2;
pr[6] = om;
pr[7] = inc;
pr[8] = phi0;
pr[9] = log(qs);

t = 7551.6; // Time at which we want to calculate the magnification

Mag = VBM.BinSourceSingleLensXallarap(pr, t); // Calculates the Binary Source magnification at time t with parameters in pr
printf("Binary Source Light Curve at time t: %lf", Mag); // Output should be 29.76...

In this function we are assuming that all properties of the sources can be deduced by their mass ratio through the mass-radius-luminosity relations specified above and customizable by the user. Therefore, the flux ratio will be FR = qs^q, where q is given by VBM.mass_luminosity_exponent and the radius of the second source will be rho * qs^p, where p is given by VBM.mass_radius_exponent.

Xallarap is also available for binary lenses through the BinSourceBinaryLensXallarap function. In this case, the parameters are 13 with the seven parameters for the static binary lens followed by the six parameters for the xallarap.

Go to Advanced control