Empirical Calibrations

Empirical Calibrations#

This page explains the empirical calibration procedures used in brutus to correct systematic errors in theoretical stellar models. For full details, see Speagle et al. (2025) §5.

Why Empirical Calibration?#

Theoretical stellar models like MIST have known systematic errors:

Temperature offsets: Models predict temperatures that differ from observations, particularly for M dwarfs (100-300 K discrepancies)
Radius discrepancies: Theoretical radii disagree with interferometric measurements, especially for low-mass stars (up to 10-20% for M dwarfs)
Photometric systematics: Synthetic photometry has band-dependent offsets due to incomplete opacities, atmospheric modeling assumptions, and photometric system differences

Without corrections, these propagate into biased distances, extinctions, and derived stellar parameters.

brutus implements two complementary corrections:

Isochrone corrections: Adjust effective temperature and radius during model generation
Photometric offsets: Multiplicative flux corrections derived from fitting results

Isochrone Corrections#

Isochrone corrections modify the predicted \(T_{\rm eff}\) and radius as a function of stellar mass. These are applied during model generation via the corr_params parameter.

Mathematical Form#

The corrections are applied to \(\log T_{\rm eff}\) and \(\log R\):

\[\Delta \log T_{\rm eff} = \log_{10}(1 + (M - 1) \times \texttt{dtdm}) \times f_{\rm EEP} \times f_{\rm [Fe/H]}\]

\[\Delta \log R = \log_{10}(1 + (M - 1) \times \texttt{drdm}) \times f_{\rm EEP} \times f_{\rm [Fe/H]}\]

where:

\(M\) is the initial mass in solar masses
\(f_{\rm EEP} = 1 - 1/(1 + \exp(-({\rm EEP} - 454)/\texttt{msto\_smooth}))\) suppresses corrections after the main-sequence turnoff
\(f_{\rm [Fe/H]} = \exp(\texttt{feh\_scale} \times {\rm [Fe/H]})\) scales corrections with metallicity
Corrections are zeroed for \(M \geq 1 M_\odot\)

Default Parameters#

The default correction parameters (from cluster calibration):

Parameter	Default	Description
`dtdm`	0.09	Temperature correction slope (dimensionless)
`drdm`	-0.09	Radius correction slope (dimensionless)
`msto_smooth`	30.0	EEP smoothing scale for turnoff suppression
`feh_scale`	0.5	Metallicity scaling factor

Usage#

Corrections are applied automatically by default. To customize:

from brutus.core import Isochrone, StellarPop

# Initialize models
iso = Isochrone()
filters = ['Gaia_G_MAW', 'Gaia_BP_MAWf', 'Gaia_RP_MAW',
           '2MASS_J', '2MASS_H', '2MASS_Ks']
pop = StellarPop(iso, filters=filters)

# Use default corrections (recommended)
seds, params, params2 = pop.get_seds(
    feh=0.0, loga=9.0, av=0.1, dist=1000.0,
    apply_corr=True  # Default
)

# Custom correction parameters
custom_corr = (0.10, -0.08, 25.0, 0.6)  # (dtdm, drdm, msto_smooth, feh_scale)
seds, params, params2 = pop.get_seds(
    feh=0.0, loga=9.0, av=0.1, dist=1000.0,
    corr_params=custom_corr
)

# Disable corrections entirely
seds_uncorr, _, _ = pop.get_seds(
    feh=0.0, loga=9.0, av=0.1, dist=1000.0,
    apply_corr=False
)

For grid generation with GridGenerator:

from brutus.core import GridGenerator, EEPTracks

tracks = EEPTracks()
filters = ['Gaia_G_MAW', 'Gaia_BP_MAWf', 'Gaia_RP_MAW',
           'PS_g', 'PS_r', 'PS_i', 'PS_z', 'PS_y']
generator = GridGenerator(tracks, filters=filters)

# Generate grid with default corrections
generator.make_grid(
    output_file='my_grid.h5',
    apply_corr=True  # Default
)

# Or with custom parameters
generator.make_grid(
    output_file='my_grid_custom.h5',
    corr_params=(0.10, -0.08, 25.0, 0.6)
)

Photometric Offsets#

Photometric offsets are multiplicative flux corrections that account for systematic differences between model predictions and observed photometry. These are computed from fitting results using bootstrap resampling.

Overview#

The photometric_offsets() function:

Takes posterior samples from BruteForce fitting
Generates model SEDs for each sample
Computes model/data flux ratios for each band
Uses likelihood reweighting to avoid circularity (for bands used in fitting)
Estimates median offsets and uncertainties via bootstrap

The key innovation is likelihood reweighting: when computing offsets for a band that was used in fitting, the function recomputes posteriors excluding that band to obtain unbiased estimates.

Basic Usage#

import numpy as np
from brutus.analysis.offsets import photometric_offsets, PhotometricOffsetsConfig

# After running BruteForce.fit() and extracting results...
# phot: observed fluxes, shape (n_objects, n_filters)
# err: flux errors, shape (n_objects, n_filters)
# mask: observation mask, shape (n_objects, n_filters)
# models: model grid coefficients
# idxs: fitted model indices, shape (n_objects, n_samples)
# reds: fitted A_V values, shape (n_objects, n_samples)
# dreds: fitted R_V values, shape (n_objects, n_samples)
# dists: fitted distances (kpc), shape (n_objects, n_samples)

# Compute offsets with default settings
offsets, errors, n_used = photometric_offsets(
    phot, err, mask, models, idxs, reds, dreds, dists
)

# Apply corrections to observed photometry
phot_corrected = phot * offsets[None, :]

Configuration#

Use PhotometricOffsetsConfig to customize behavior:

config = PhotometricOffsetsConfig(
    min_bands_used=5,        # Minimum bands for reweighted objects
    min_bands_unused=3,      # Minimum bands for non-reweighted objects
    n_bootstrap=500,         # Bootstrap iterations
    uncertainty_method='bootstrap_iqr',  # 'bootstrap_iqr' or 'bootstrap_std'
    random_seed=42           # For reproducibility
)

offsets, errors, n_used = photometric_offsets(
    phot, err, mask, models, idxs, reds, dreds, dists,
    config=config
)

Additional Parameters#

offsets, errors, n_used = photometric_offsets(
    phot, err, mask, models, idxs, reds, dreds, dists,
    sel=quality_mask,           # Boolean selection of objects to use
    weights=sample_weights,     # Per-sample weights
    mask_fit=fitting_mask,      # Which filters were used in fitting
    old_offsets=previous_offs,  # Remove previous offsets before recomputing
    dim_prior=True,             # Apply dimensionality prior in reweighting
    prior_mean=prior_offs,      # Gaussian prior means
    prior_std=prior_errs,       # Gaussian prior standard deviations
    verbose=True
)

Interpreting Results#

The function returns:

offsets: Multiplicative flux corrections (model/data ratios). Apply as phot_corrected = phot * offsets
errors: Uncertainties from bootstrap distribution
n_used: Number of objects used per filter (useful for quality assessment)

Offsets near 1.0 indicate good agreement between models and data. Values significantly different from 1.0 suggest systematic calibration differences.

Pre-Computed Offsets#

The default brutus grids include empirically calibrated offsets derived from:

Open cluster fitting: Six benchmark clusters (including M67) used to derive isochrone corrections and initial photometric offsets
Field star validation: ~23,000 nearby, low-reddening stars with precise Gaia parallaxes

See Speagle et al. (2025) §5.2-5.3 for calibration details and Table 5 for the derived offset values.

When to Use Custom Calibrations#

Use the default calibrations for most applications. Consider custom calibrations when:

Working with photometric systems not in the default set
Analyzing populations very different from the calibration sample (e.g., very metal-poor)
Combining data from surveys with known cross-calibration issues

Limitations#

Metallicity range: Calibrations are optimized for near-solar metallicity. Metal-poor stars may have different systematics.
Evolutionary phase: Corrections are derived primarily from main-sequence stars. Giants and other phases may require different treatment.
Survey-specific: Each photometric system has unique calibration issues. Offsets don’t transfer between surveys.

References#

Speagle et al. (2025), “Deriving Stellar Properties, Distances, and Reddenings using Photometry and Astrometry with BRUTUS”, arXiv:2503.02227 (see §5 for calibration details)