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.

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*:

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

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

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

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

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

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

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.

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.

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).

If you have installed it successfully, just type

jktld

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.

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.

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).

- JKTLD does simple linear interpolation in
*Teff*and*logg*. Thus it is restricted to specific values of*[M/H]*and*Vmicro*as and when they are available in the original tables. Given the extremely limited coverage of*[M/H]*and*Vmicro*by the available tabulations of limb darkening coefficients, I have no plans to modify this. - The limb darkening coefficients in Claret (2000) and Claret (2004) from the Phoenix model atmospheres cover
*Teff*s between 2000 and 9800 K. However, there is a gap in coverage between 5800 and 7000 K because model atmospheres were not available for these*Teff*s. This means that the relevant coefficients outputted by JKTLD are interpolated between two quite different*Teff*s so are a bit less reliable. The ATLAS9 coefficients from the same papers are not affected.

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