meidem 1.0.3

Multi-grid Epic Interpolator for limb DarkEning Models

MEIDEM - Multi-grid Epic Interpolator for stellar limb DarkEning Models

MEIDEM is a Python package that provides a unified interface for interpolating stellar limb darkening (LD) coefficients from multiple published grids. Instead of dealing with different table formats, column names, and interpolation strategies for each grid, you call a single function — get_ld_coefficients() — and MEIDEM handles the rest.

All grid tables are bundled with the package: no manual downloads, no VizieR visits, no path configuration.

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What is Stellar Limb Darkening?

When a planet transits its host star, the transit light curve is not only shaped by the planet's size and orbital geometry — it is also strongly affected by how the stellar brightness varies across the disk. Stars are not uniformly bright: they are brighter at the center and dimmer toward the edge (the "limb"). This effect, called limb darkening, must be correctly accounted for to accurately determine the planetary radius and other transit parameters.

Limb darkening is typically described by a mathematical law parametrized by one or more coefficients that depend on the stellar type (Teff, logg, metallicity) and the photometric bandpass. These coefficients are computed from model stellar atmospheres and published as tabulated grids. MEIDEM interpolates those grids for any given set of stellar parameters.

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Installation

pip install meidem

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Quick Start

import meidem

# Get power-2 LD coefficients for a Sun-like star observed with TESS
result = meidem.get_ld_coefficients(
    teff=5778,        # effective temperature (K)
    logg=4.44,        # surface gravity (cgs)
    feh=0.0,          # metallicity [Fe/H] (dex)
    passband='TESS',
    grid='kostogryz2022',
    law='power2',
)

print(result['coefficients'])   # [c, alpha]
print(result['reference'])      # full bibliographic reference
print(result['doi'])            # DOI of the source paper

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Solar Reference Example (IAU 2015 Values)

The example below uses the IAU 2015 nominal solar parameters as a sanity check. You can use it to verify the installation and compare results across grids.

import meidem

# IAU 2015 nominal solar parameters
# Teff = 5772 K  (Prša et al. 2016, AJ 152, 41)
# logg = 4.438   (log10 of GM_sun / R_sun^2 in cgs)
# [Fe/H] = 0.0   (by definition)
SUN = dict(teff=5772, logg=4.438, feh=0.0)

print("LD coefficients for the Sun (IAU 2015)\n")

# Kostogryz+2022 — power-2 for TESS
r = meidem.get_ld_coefficients(**SUN, passband='TESS',
                                grid='kostogryz2022', law='power2')
print(f"Kostogryz+2022  power-2    TESS   : {r['coefficients']}")
# Expected: [c ≈ 0.65, alpha ≈ 0.60]

# Kostogryz+2022 — nonlinear (4-coeff) for TESS
r = meidem.get_ld_coefficients(**SUN, passband='TESS',
                                grid='kostogryz2022', law='nonlinear')
print(f"Kostogryz+2022  nonlinear  TESS   : {r['coefficients']}")

# Claret & Southworth 2022 — power-2 for TESS
r = meidem.get_ld_coefficients(**SUN, passband='TESS', grid='claret2022')
print(f"Claret2022      power-2    TESS   : {r['coefficients']}")

# Claret 2017 — quadratic for TESS, ATLAS model, Least-Squares
r = meidem.get_ld_coefficients(**SUN, passband='TESS',
                                grid='claret2017', law='quadratic',
                                mod='A', met='L')
print(f"Claret2017      quadratic  TESS   : {r['coefficients']}")

# Claret & Bloemen 2011 — quadratic for Kepler
r = meidem.get_ld_coefficients(**SUN, passband='Kp',
                                grid='claret2011', law='quadratic')
print(f"Claret2011      quadratic  Kepler : {r['coefficients']}")

Loading your own star table

import pandas as pd
import meidem

# Load a table with stellar parameters
df = pd.read_csv('my_stars.csv')
# Required columns: Teff, logg, FeH

results = []
for _, row in df.iterrows():
    try:
        r = meidem.get_ld_coefficients(
            teff=row['Teff'],
            logg=row['logg'],
            feh=row['FeH'],
            passband='TESS',
            grid='kostogryz2022',
            law='power2',
        )
        results.append({
            'star' : row['hostname'],
            'LD_c1': r['coefficients'][0],
            'LD_c2': r['coefficients'][1],
            'status': 'OK',
        })
    except ValueError:
        results.append({'star': row['hostname'], 'status': 'OUT_OF_GRID'})

df_ld = pd.DataFrame(results)
print(df_ld)

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Supported Grids

MEIDEM currently supports four published grids. Each covers a different combination of atmospheric models, LD laws, and photometric passbands.

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1. kostogryz2022 — Kostogryz et al. (2022), MPS-ATLAS

Reference: Kostogryz, N. V. et al. (2022), A&A — doi:10.1051/0004-6361/202140376

Atmospheric model: MPS-ATLAS — a modern 1D plane-parallel model developed at the Max Planck Institute for Solar System Research. It covers a wide range of metallicities and is one of the most up-to-date grids available for exoplanet transit modeling.

Passbands: TESS, Kepler, CHEOPS, PLATO

Grid limits: Teff ≈ 3500–10000 K · logg ≈ 2.0–5.0 · [Fe/H] ≈ −5.0 to +1.0

Laws available:

Law key	Coefficients	Equation
power2	c, alpha	I(μ)/I(1) = 1 − c(1 − μ^α)
nonlinear	a1, a2, a3, a4	I(μ)/I(1) = 1 − Σ aₙ(1 − μ^(n/2))

When to use: This is the recommended grid for TESS and Kepler as of 2022. Use law='power2' for most transit fitting applications — the power-2 law shows the lowest residuals and the smallest bias in recovered Rp/R* compared to the quadratic law for TESS photometry (Maxted 2023). Note that stars cooler than ~3500 K (cool M dwarfs) fall outside this grid.

r = meidem.get_ld_coefficients(
    teff=5778, logg=4.44, feh=0.0,
    passband='TESS',
    grid='kostogryz2022',
    law='power2',       # or 'nonlinear'
)

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2. claret2022 — Claret & Southworth (2022), A&A 664, A128

Reference: Claret, A. & Southworth, J. (2022), A&A 664, A128 — doi:10.1051/0004-6361/202244278

Atmospheric model: ATLAS (plane-parallel, 1D).

Passbands: TESS, Kepler, Gaia (G, BP, RP), SDSS (u, g, r, i, z), Johnson (U, B, V, R, I), 2MASS (J, H, K), Strömgren (u, v, b, y)

Grid limits: Teff ≈ 3500–50000 K · logg ≈ 0.0–5.0 · [Fe/H] ≈ −5.0 to +1.0

Laws available: Power-2 only

Microturbulence (xi): Available at ξ = 0, 1, 2, 4, 8 km/s. Default is ξ = 2.0 km/s, appropriate for most FGK main-sequence stars.

When to use: Excellent choice when you need power-2 coefficients for a wide range of passbands beyond TESS and Kepler — especially Gaia, SDSS, or Johnson bands. Also useful for comparison with kostogryz2022 for TESS and Kepler.

r = meidem.get_ld_coefficients(
    teff=5778, logg=4.44, feh=0.0,
    passband='Gaia_G',   # or 'TESS', 'Kepler', 'SDSS_r', 'Johnson_V', ...
    grid='claret2022',
    xi=2.0,
)

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3. claret2017 — Claret (2017), A&A 600, A30

Reference: Claret, A. (2017), A&A 600, A30 — doi:10.1051/0004-6361/201629705

Atmospheric models: ATLAS and PHOENIX

Passbands: TESS only

Grid limits (ATLAS): Teff ≈ 3500–50000 K · logg ≈ 0.0–5.0 · [Fe/H] ≈ −5.0 to +1.0 · ξ = 0–8 km/s

Grid limits (PHOENIX): Teff ≈ 1500–12000 K · logg ≈ 2.5–6.0 · [Fe/H] = 0.0 only

Laws available:

Law key	N coeff	Notes
linear	1	Avoid — produces biases up to ~3% in Rp/R* (Espinoza & Jordán 2016)
quadratic	2	Most widely used; fast computation (Kopal 1950; Kreidberg 2015)
square-root	2	Better for hot stars
logarithmic	2	Better performance than quadratic (Espinoza & Jordán 2016)
4coeff	4	Most flexible; best physical description of intensity profile
y	1	Gravity darkening coefficient; ATLAS only

Fitting methods (met):

Key	Method	When to use
'L'	LSM — Least-Squares	Minimizes residuals to the intensity profile. Most common; default.
'F'	PCM/CFM — Flux Conservation	Ensures integrated flux is conserved. Preferred for generating synthetic light curves.

Atmospheric models (mod):

Key	Model	When to use
'A'	ATLAS	Covers all metallicities; default for most stars
'P'	PHOENIX	More realistic for cool stars (Teff < 5000 K); [Fe/H] = 0.0 only

When to use: This grid is ideal when you want to compare multiple LD laws for TESS — for example, to assess how much the choice of law affects the recovered Rp/R*. The 4coeff law provides the most accurate physical description; quadratic is most compatible with transit fitting codes like batman.

# ATLAS — quadratic, Least-Squares (most common setup)
r = meidem.get_ld_coefficients(
    teff=5778, logg=4.44, feh=0.0,
    passband='TESS',
    grid='claret2017',
    law='quadratic',
    mod='A',    # 'A' = ATLAS | 'P' = PHOENIX
    met='L',    # 'L' = Least-Squares | 'F' = Flux Conservation
    xi=2.0,
)

# PHOENIX — better for cool stars (Teff < 5000 K), [Fe/H] = 0.0 only
r = meidem.get_ld_coefficients(
    teff=3800, logg=4.8, feh=0.0,
    passband='TESS',
    grid='claret2017',
    law='quadratic',
    mod='P',
)

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4. claret2011 — Claret & Bloemen (2011), A&A 529, A75

Reference: Claret, A. & Bloemen, S. (2011), A&A 529, A75 — doi:10.1051/0004-6361/201116451

Atmospheric models: ATLAS and PHOENIX

Passbands: Kepler (Kp), CoRoT (C), Spitzer ch1 (S1), Spitzer ch2 (S2)

Laws available: quadratic, root-square, logarithmic, 4coeff, linear, y

Fitting methods (met) and models (mod): Same as Claret 2017.

When to use: This is the standard reference grid for Kepler and CoRoT — one of the most cited LD grids in the exoplanet literature. If you are working with Kepler light curves, this is the grid to use for comparison with published results.

r = meidem.get_ld_coefficients(
    teff=5778, logg=4.44, feh=0.0,
    passband='Kp',          # 'Kp' = Kepler | 'C' = CoRoT | 'S1'/'S2' = Spitzer
    grid='claret2011',
    law='quadratic',
    mod='A',
    met='L',
    xi=2.0,
)

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Choosing the Right Grid and Law

Which grid?

Situation	Recommended
TESS photometry, modern analysis	kostogryz2022
Kepler photometry	claret2011 or kostogryz2022
Gaia, SDSS, Johnson, 2MASS passbands	claret2022
Cool stars (Teff < 3500 K) with TESS	claret2017 with mod='P'
Comparing grids (e.g. for a paper)	kostogryz2022 + claret2022

Which law?

The power-2 law shows the lowest residuals and the smallest bias in recovered transit parameters for TESS and Kepler photometry. The quadratic law is the most widely used in the exoplanet literature because it enables fast and efficient light curve computation in popular codes such as batman (Kreidberg 2015). Laws such as logarithmic, square-root, and 3-parameter do a better job than quadratic and linear when deriving transit parameters (Espinoza & Jordán 2016).

Law	Recommendation
power2	Best overall for TESS/Kepler. 2 parameters, excellent accuracy (Maxted 2023).
quadratic	Most compatible with transit codes. Fast, 2 parameters, widely tabulated.
nonlinear / 4coeff	Best physical description. Ideal when fixing coefficients rather than fitting them.
logarithmic / square-root	Better than quadratic, less widely supported by transit codes.
linear	Avoid — biases up to ~3% in Rp/R* (Espinoza & Jordán 2016).

Grid coverage and known limitations

Each grid covers a finite region of stellar parameter space. Stars outside the grid raise a ValueError — MEIDEM never extrapolates silently.

Grid	Teff range (K)	ATLAS	PHOENIX
kostogryz2022	~3500–10000	✓	—
claret2022	~3500–50000	✓	—
claret2017	~3500–50000 / ~1500–12000	✓	✓ ([Fe/H]=0 only)
claret2011	~3500–50000 / ~1500–12000	✓	✓ ([Fe/H]=0 only)

Note for cool M dwarfs (Teff < 5000 K): For stars cooler than about 5000 K, significant deviations between measured and theoretical LDCs (Δu₁, Δu₂ ≈ 0.2) have been observed in TESS light curves (Patel & Espinoza 2022). Treat results with extra caution for very cool stars.

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API Reference

meidem.get_ld_coefficients()

meidem.get_ld_coefficients(
    teff,                       # float — effective temperature (K)
    logg,                       # float — surface gravity log g (cgs)
    feh,                        # float — metallicity [Fe/H] (dex)
    passband,                   # str   — photometric passband
    grid    = 'kostogryz2022',  # str   — grid key
    law     = None,             # str   — LD law (None = grid default)
    xi      = 2.0,              # float — microturbulence km/s (ATLAS only)
    met     = 'L',              # str   — fitting method: 'L' or 'F'
    mod     = 'A',              # str   — atmospheric model: 'A' or 'P'
    verbose = False,            # bool  — print grid info
)

Returns:

{
    'coefficients': [c, alpha],       # interpolated LD coefficients
    'n_coeffs'    : 2,                # number of coefficients
    'law'         : 'power2',         # LD law used
    'passband'    : 'TESS',           # passband used
    'grid'        : 'kostogryz2022',  # grid used
    'reference'   : 'Kostogryz et al. (2022)',
    'doi'         : '10.1051/0004-6361/202140376',
    'teff_input'  : 5778.0,
    'logg_input'  : 4.44,
    'feh_input'   : 0.0,
    'xi'          : None,   # float if Claret ATLAS, else None
    'met'         : None,   # str if Claret 2017/2011, else None
    'mod'         : None,   # str if Claret 2017/2011, else None
}

Raises ValueError if parameters are outside grid limits.

Discovery functions

meidem.available_grids()                  # all grids with details
meidem.available_laws('claret2017')       # laws for a grid
meidem.available_passbands('claret2022')  # passbands for a grid

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Citation

If you use MEIDEM in your research, please cite the package and the grid(s) you used:

@software{meidem,
  author  = {Meidem, Icaro},
  title   = {{MEIDEM}: Multi-grid Exoplanet Interpolator for limb DarkEning Models},
  year    = {2026},
  url     = {https://github.com/icaromeidem/meidem},
}

Grid references:

• Kostogryz et al. (2022), A&A — doi:10.1051/0004-6361/202140376
• Claret & Southworth (2022), A&A 664, A128 — doi:10.1051/0004-6361/202244278
• Claret (2017), A&A 600, A30 — doi:10.1051/0004-6361/201629705
• Claret & Bloemen (2011), A&A 529, A75 — doi:10.1051/0004-6361/201116451

Key references used in this documentation:

• Espinoza & Jordán (2015), MNRAS 450, 1879
• Espinoza & Jordán (2016), MNRAS 457, 3573
• Maxted (2023), MNRAS 519, 3723
• Patel & Espinoza (2022), AJ 163, 228

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License

MIT © Icaro Meidem