JKTLD - for calculating limb darkening coefficients


Limb darkening is a natural property of stars. The phenomenon is that the surface brightness is lower if you look at the surface of a star from an angle rather than from directly above. This means that if you see a star in the sky, the edge ("limb") of the star is fainter than the centre. This can be seen in the Sun, either through projecting its light using a telescope or from looking through moderately thin cloud (be careful!). Limb darkening (LD) is important in several areas of stellar physics, including the study of transiting extrasolar planets, eclipsing binary stars, interferometry and some gravitational microlenses.

Many observations of star require the strength of LD to be estimated, which can be done using theoretical models of stellar atmospheres. JKTLD is a tool to help in these circumstances. JKTLD outputs theoretically-calculated LD strengths for a number of LD laws: equations which predict the amount of LD as a function of which bit of a star you are looking at. The coefficients of these laws are obtained by bilinear interpolation (in effective temperature and surface gravity) in published tables of coefficients calculated from stellar model atmospheres by several researchers.


Choice of limb darkening laws

Stellar limb darkening (LD) laws are specified as a function of μ = cos γ where γ is the angle between a line normal to the stellar surface and the line of sight of the observer.

The linear LD law can be traced back to Schwarzschild (1906, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, p. 43). It has one LD coefficient, c:

[linear limb darkening law]

The quadratic LD law is due to Kopal (1950, Harvard Col. Obs. Circ., 454, 1) and has two LD coefficients:

[quadratic limb darkening law]

The square-root LD law (Díaz-Cordovés & Giménez, 1992, A&A, 259, 227) is:

[square-root limb darkening law]

The logarithmic LD law (Klinglesmith & Sobieski, 1970, AJ, 75, 175) is:

[logarithmic limb darkening law]

The exponential LD law (Claret & Hauschildt, 2003, A&A, 412, 241) is:

[exponential limb darkening law]

The Sing three-parameter LD law (Sing et al., 2009, A&A, 505, 891) is:

[three-par limb darkening law]

Finally, the Claret four-parameter LD law (Claret, 2000, A&A, 363, 1081) is:

[four-par limb darkening law]


Sources of limb darkening coefficient tables

JKTLD interpolates within large tables of results calculated in several theoretical studies. These studies are:

Study Model atmosphere Teff range logg range [M/H] range Vmicro range
Van Hamme (1993, AJ, 106, 2096) ATLAS9 3500 to 50000 0.0 to 5.0 0.0 n/a
Díaz-Cordovés et al (1995, A&AS, 110, 329) ATLAS9 3500 to 50000 0.0 to 5.0 0.0 n/a
Claret et al (1995, A&AS, 114, 247) ATLAS9 3500 to 50000 0.0 to 5.0 0.0 n/a
Claret (2000, A&A, 363, 1081) ATLAS9 3500 to 50000 0.0 to 5.0 -5.0 to +0.5 0,1,2,4,8
Claret (2000, A&A, 363, 1081) Phoenix 2000 to 9800 3.5 to 5.5 0.0 2
Claret & Hauschildt (2003, A&A, 412, 241) Phoenix 5000 to 10000 0.0 to 5.0 0.0 2
Claret (2004, A&A, 428, 1001) ATLAS9 3500 to 50000 0.0 to 5.0 -5.0 to +1.0 0,1,2,4,8
Claret (2004, A&A, 428, 1001) Phoenix 2000 to 9800 3.5 to 5.0 0.0 2
Sing (2010, A&A, 510, A21) ATLAS9 3500 to 50000 0.0 to 5.0 -5.0 to +1.0 n/a


Most studies do not cover all the available LD laws or photometric passbands:

Study lin quad log sqrt exp 3par 4par Passbands
Van Hamme (1993, AJ, 106, 2096) bolometric, Strömgren uvby, Johnson UBVRIJHKLMN, Cousins RI
Díaz-Cordovés et al (1995, A&AS, 110, 329) Strömgren uvby, Johnson UBV
Claret et al (1995, A&AS, 114, 247) Cousins RI, Teide JHK
Claret (2000, A&A, 363, 1081) bolometric, Strömgren uvby, Johnson UBV, Cousins RI, Teide JHK
Claret & Hauschildt (2003, A&A, 412, 241) Strömgren uvby, Johnson UBV, Cousins RI, Teide JHK
Claret (2004, A&A, 428, 1001) SDSS ugriz
Sing (2010, A&A, 510, A21) CoRoT (white) and Kepler satellite passbands

If you want LD coefficients calculated for individual CCDs on board the Kepler satellite then these can be obtained from Andrej Prsa. I do not include these in JKTLD for two reasons. Firstly, to avoid a large increase in the complexity of the program and the size of the set of datafiles it uses. Secondly, because I do not think this level of detail is necessary given that the LD coefficients are based on theoretical calculations so only approximately represent reality anyway.


Obtaining JKTLD

The source code for JKTLD is a monolithic program written in practically standard FORTRAN 77. You can obtain version 3 (2010/12/06) from here. It also requires a set of datafiles containing LD tabulations in a standard format. For convenience I have packaged the source code and all the LD tables into one tarfile which can be obtained here. For reference the previous version 2 of the code can be obtained from here.


Compiling JKTLD

Because JKTLD is a single FORTRAN 77 program, it is relatively easy to compile. These instructions are for the Unix/Linux operating systems.

Firstly, unpack the tarfile into a new directory. Then put the directory path and filenames into the "data FILES /" statement on line 128, replacing the "/home/jkt/tables/...." lines which are for my own filesystem. Then compile JKTLD with the compiler of your choice into an executable file (I call mine "jktld"). Place the executable file into your "bin" directory or any other directory in your $PATH.

JKTLD was originally written using the g77 compiler, and this should still compile it successfully. It now works with the g95 and gfortran compilers running on my laptop (IBM Thinkpad with kubuntu 10.04) and with the Intel ifort compiler running under Scientific Linux. Please let me know if you have trouble compiling or running the code (my email address is on my homepage).


Running JKTLD

If you have installed it successfully, just type


and it will output some instructions about the two options for obtaining LD coefficients. The two options require different numbers of arguments at the command line.


Running JKTLD to obtain an output file

If you want a large number of LD coefficients, covering different laws and passbands, this option is for you. Usage:

jktld  [Teff]  [logg]  [M/H]  [Vmicro]  [outfile]

so for example you could type:

jktld 5743 4.23 0.0 2 jktld.out

and it will return an output file called "jktld.out" which looks like this. The output file contains some important notes and interpolated LD coefficients for a star with Teff = 5743 K, logg = 4.23 (c.g.s), [M/H] = 0.0 and Vmicro = 2. The output file includes LD coefficients for all available passbands and for most of the LD laws.

[M/H] is the atmospheric metallicity. Its possible values are -5.0, -4.5, -4.0, -3.5, -3.0, -2.5, -2.0, -1.5, -1.0, -0.5, -0.3, -0.2, -0.1, 0.0, 0.1, 0.2, 0.3, 0.5 and 1.0. Most LD coefficients are only available for [M/H] = 0.0.

Vmicro is the microturbulence velocity in km/s. Its possible values are 0, 1, 2, 4 and 8 km/s. Most coefficients are only available for Vmicro = 2.


Running JKTLD to obtain coefficients to standard output

If you want just a few LD coefficients, or if you want ones for the exponential or four-parameter LD laws, then you can run JKTLD using additional command-line arguments and it will return the coefficients you want to standard output. Usage:

jktld  [Teff]  [logg]  [M/H]  [Vmicro]  [law]  [table]  [filter]

so for example you could type:

jktld 5743 4.23 0.0 2 f 6 Sr

and you should see the following coefficients appear at the terminal:

    0.5955    -0.1625     0.7276    -0.3953

The [law] value indicates which LD law you want the coefficients for and can be either u (linear), q (quadratic), l (logarithmic), s (square root), e (exponential), t (Sing 3-parameter) or f (Claret 4-parameter).

The [table] value must be an integer between 1 and 8. Type 1 for Van Hamme (1993), 2 for Díaz-Cordovés et al (1995) and Claret et al (1995), 3 for Claret (2000) ATLAS9, 4 for Claret (2000) Phoenix, 5 for Claret & Hauschildt (2003), 6 for Claret (2004) ATLAS9, 7 for Claret (2004) Phoenix, or 8 for Sing (2010).

The [filter] value is a two-character code for the passband you want. Choose from: bo (bolometric), uS, vS, bS, yS (Strömgren), UJ, BJ, VJ, RJ, IJ, JJ, HJ, KJ, LJ, MJ, NJ (Johnson), Su, Sg, Sr, Si, Sz (SDSS), Co (CoRoT satellite white-light), or Kp (Kepler satellite).




Last modified: 2013/02/18           John Southworth   (Keele University, UK)