LimbDarkening.md

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Limb darkening

Real stars are not uniform disks. The intensity declines from the center to the edge of the source, as described by limb darkening laws. VBMicrolensing has several built-in limb darkening laws and allows the user to specify an arbitrary law.

Linear limb darkening

The default limb darkening profile is linear, according to

$$ I(\nu) = I(0) \left[1 - a1 (1 - \nu)\right] $$

with $\nu=\sqrt{1-r^2/\rho^2}$.

The coefficient VBM.a1 specifies the linear limb darkening coefficient. We note that there are other popular expressions for linear limb darkening that can be reduced to the one adopted in VBMicrolensing through simple transformations (e.g. for $\Gamma_1$ see An et al. ApJ 572:521 (2002), Eq. (11)).

In order to use linear limb darkening in all calculations with extended sources in VBMicrolensing, it is sufficient to give a non-vanishing value to VBM.a1:

VBM.a1 = 0.51; // Linear limb darkening coefficient. 
Mag = VBM.BinaryMag2(s, q, y1, y2, Rs); // Call to the BinaryMag2 with the same parameter as in the previous section
printf("Magnification with limb darkened source = %lf\n", Mag);  // Output should be 18.27.....

In order to go back to uniform sources, just set VBM.a1 = 0. In general, a calculation with limb darkening is slower than a calculation with a uniform source, because it is repeated on several concentric annuli.

There are three more limb darkening laws that are already available in VBMicrolensing. You may switch from one to another using the function VBM.SetLDprofile.

If you want to go back to linear limb darkening, you may use VBM.SetLDprofile(VBM.LDlinear);

Square root limb darkening

$$I(\nu) = I(0) \left[1 - a1 (1 - \nu)- a2 (1-\sqrt{\nu}) \right]$$

This law has two parameters that are given through VBM.a1 and VBM.a2:

VBM.SetLDprofile(VBM.LDsquareroot); 
VBM.a1 = 0.51;
VBM.a2 = 0.3;
Mag = VBM.BinaryMag2(s, q, y1, y2, Rs);
printf("Magnification with square root limb darkened source = %lf\n", Mag);  // Output should be 18.2730.....

Quadratic limb darkening

$$I(\nu) = I(0) \left[1 - a1 (1 - \nu)- a2 (1-\nu)^2 \right]$$

VBM.SetLDprofile(VBM.LDquadratic); 
VBM.a1 = 0.51;
VBM.a2 = 0.3;
Mag = VBM.BinaryMag2(s, q, y1, y2, Rs);
printf("Magnification with quadratic limb darkened source = %lf\n", Mag);  // Output should be 18.2728.....

Logarithmic limb darkening

$$I(\nu) = I(0) \left[ 1 - a1 (1 - \nu)- a2 ~ \nu \ln{\nu} \right]$$

VBM.SetLDprofile(VBM.LDlog); 
VBM.a1 = 0.51;
VBM.a2 = 0.3;
Mag = VBM.BinaryMag2(s, q, y1, y2, Rs);
printf("Magnification with logarithmic limb darkened source = %lf\n", Mag);  // Output should be 18.2788.....

User-defined limb darkening

Suppose you have your favorite limb darkening law. Then you should define a function with the syntax double MyLDprofile(double r);

If there are any parameters, they must be defined as external variables accessible in the scope in which you use your profile. For example, you may define them as global variables. Here is an example of a valid limb darkening function:

double MyLDprofile(double r) {
	double costh;
	costh = sqrt(1 - r * r);
	return (1 - u1 * (1 - costh) - u2 * (1 - sqrt(costh))); // square root
}

This is just a square root limb darkening law depending on two parameters u1 and u2, which must be defined as global.

At this point, in order to adopt this user-defined function as the current limb darkening law, you must still use the VBM.SetLDprofile function as follows:

u1 = 0.51;
u2 = 0.3;
VBM.SetLDprofile(&MyLDprofile, 1000); // The limb darkening law is pre-calculated on a grid of 1000 points.
Mag = VBM.BinaryMag2(s, q, y1, y2, Rs);
printf("Magnification with user-defined LD profile = %lf\n", Mag);  // Output should be 18.2712.....

Note that the two-parameters version of VBM.SetLDprofile requires the location of the function as the first parameter and the number of points in the grid as the second parameter. The limb darkening law is then pre-calculated on this grid in order to perform all subsequent extended source calculations.

Warning: the user-defined limb darkening is not available in the Python version of VBMicrolensing.

Multi-band observations

Sometimes, simultaneous observations in multi-band are available. In this case, it is possible to repeat the calculation of the magnification for each band changing VBM.a1 as appropriate.

However, there is the possibility to save some time by BinaryMagMultiDark. This function exploits the same sequence of concentric annuli to calculate the magnification with different limb darkening coefficients simultaneously. The computational time is then comparable to a single calculation. Here is an example:

const int nbands=4; // number of bands
double a1_list[nbands] = { 0.2,0.3,0.51,0.6 }; // list of limb darkening coefficients
double mag_list[nbands]; // array where to store the output
double Tol = 1.e-3; // Accuracy (see section about accuracy control)
VBM.BinaryMagMultiDark(s, q, y1, y2, rho, a1_list, nbands, mag_list, Tol);
for (int i = 0; i < 4; i++) {
   printf("BinaryMagMultiDark at band %d: %lf\n", i, mag_list[i]);
}

BinaryMagMultiDark currently works with linear limb darkening only. It is a particular function that has been introduced after a specific scientific request, but we believe it can be useful to general users with easy customization if necessary.

Go to: Accuracy control