How brutus works

brutus infers the distance, extinction and properties of a star from its photometry and, if available, its parallax [Speagle et al., 2025]. It does not run a sampler. It evaluates every model in a grid of stellar models, integrates the priors for each one, and builds the posterior as a weighted sum over models.

The four steps of a brutus fitEvery grid model is fitted for distance and dust; the good fits are kept; for each, Monte Carlo draws in log-distance and dust are weighted by the priors; the posterior is the weighted sum over models.1 Fit every modelmass, age, metallicitybest distance and dust each2 Keep the good fitswithin 10⁻³ of the best3 Integrate the priorslog distance →draws in pairs (z, −z),weighted by the priors4 Draw the posteriordistance →the sum over models;250 draws are saved

The four steps of a fit, described in The fit in four steps.

The forward model

A model star has intrinsic parameters \(\theta\), which set its spectral energy distribution (SED), and extrinsic parameters \(\phi\), which set how it appears to us:

Parameter

Meaning

\(\theta\)

\(M_{\rm init}\)

initial mass, in \(M_\odot\)

EEP

equivalent evolutionary point: the position along the evolutionary track

[Fe/H]

metallicity

\(\phi\)

\(d\)

distance

\(A_V\)

extinction in the V band, in mag

\(R_V\)

shape of the extinction curve

Age is not a parameter: it follows from \((M_{\rm init}, {\rm EEP}, {\rm [Fe/H]})\). Stellar models explains the tracks and why EEP is used.

Each grid model stores its magnitudes \(M_b(\theta)\) in every band \(b\) at a distance of 1 kpc, and two reddening vectors \(R_b(\theta)\) and \(R'_b(\theta)\). The predicted flux in band \(b\) is

\[ F_b(\theta, \phi) = s \; 10^{-0.4\,\left[ M_b(\theta) + A_V \left( R_b(\theta) + R_V\, R'_b(\theta) \right) \right]}, \qquad s = \left( \frac{1\,{\rm kpc}}{d} \right)^2 , \]

where \(s\) is the scale factor. Distance dims all bands by the same factor; dust dims blue bands more than red ones. The extinction curve follows Fitzpatrick and Massa [2009], with \(R_V\) as its shape parameter.

The forward modelA model SED at 1 kpc is scaled to the distance of the star and dimmed by dust, which dims blue bands more than red ones, and compared with the observed photometry.wavelengthbrightnessgrizyJHKsmodel SED at 1 kpcmoved to distance ddimmed by dust; datascale = (1 kpc / d)²dust: most in blue

The likelihood

brutus compares fluxes, not magnitudes, because flux errors are close to Gaussian and magnitude errors are not. For the observed fluxes \(\hat F_b\) with errors \(\sigma_b\), over the \(N_b\) valid bands,

\[ \chi^2(\theta, \phi) = \sum_{b=1}^{N_b} \frac{\left( \hat F_b - F_b(\theta, \phi) \right)^2}{\sigma_b^2}, \qquad \ln \mathcal{L}_{\rm phot} = -\tfrac{1}{2}\,\chi^2 + \text{normalization}. \]

The normalization holds the Gaussian terms in \(\sigma_b\) and, by default, a term in the number of bands (logl_dim_prior), so that stars with different numbers of bands are weighted consistently.

A parallax \(\hat\varpi \pm \sigma_\varpi\), in mas, adds a Gaussian term in \(\varpi = 1/d\) with \(d\) in kpc:

\[ -2 \ln \mathcal{L}_\varpi(d) = \frac{\left( \hat\varpi - 1/d \right)^2}{\sigma_\varpi^2} + {\rm const}. \]

A fit needs at least 4 bands: each model has 3 free parameters (\(s\), \(A_V\), \(R_V\)).

The priors

The model prior depends only on the grid model; the position-dependent priors depend on where the star is; the \(R_V\) prior is the same for all stars.

\[ \pi(\theta, \phi) = \underbrace{\pi_{\rm IMF}(M_{\rm init}) \; \frac{dt}{d\,{\rm EEP}} \; \Delta V(\theta)}_{\text{model prior}} \;\times\; \underbrace{\pi_{\rm gal}(d, \theta \mid \ell, b) \; \pi_{\rm dust}(A_V \mid d, \ell, b)}_{\text{position-dependent priors}} \;\times\; \pi(R_V) \]

  • Initial mass function: the Kroupa [2001] IMF.

  • Age weight \(dt/d\,{\rm EEP}\): the time a star spends per unit of EEP. It turns a prior uniform in EEP into one uniform in age.

  • Grid volume \(\Delta V\): the spacing of the grid in each parameter, so that a sum over models approximates an integral. The default grid steps EEP by 6 on the main sequence and by 2 after it.

  • Galactic structure \(\pi_{\rm gal}\): the density of a thin disk, a thick disk and a halo along the sightline \((\ell, b)\), each with its own metallicity distribution and an age distribution set by that metallicity.

  • Dust map \(\pi_{\rm dust}\): the Bayestar 3D dust map [Green et al., 2019], if given, with 0.2 mag of scatter added to the map’s own spread. Where the map has no reliable value, this prior is uniform in \(A_V\).

  • Extinction curve \(\pi(R_V)\): Gaussian with mean 3.32 and standard deviation 0.18 [Schlafly et al., 2016]. It enters the fit of each model in step 1, and so the Gaussian that step 3 draws from.

Priors gives each prior’s equation, defaults and function.

The fit in four steps

1. Fit every model. For each grid model, brutus finds the \((s, A_V, R_V)\) that maximize \(\mathcal{L}_{\rm phot}\,\pi(R_V)\) by Gauss-Newton iteration: first in magnitudes for every model, then in fluxes for the models whose magnitude-space fit is close to the best. It also computes their 3×3 Fisher matrix, which describes a Gaussian around the best fit.

2. Keep the good fits. Each model gets the weight \(\mathcal{L}_{\rm phot}\,\pi_{\rm model}\). If a parallax is given, the weight is multiplied by the parallax likelihood averaged over the model’s Gaussian in \(\ln d\): \(\int \mathcal{N}(\ln d)\,\mathcal{N}(\varpi \mid 1/d,\,\sigma_\varpi)\,d\ln d\). Its width is the uncertainty of \(\ln d\) after marginalizing over \(A_V\) and \(R_V\). A five-node Gauss-Hermite rule, centred on the peak of the integrand, evaluates it for every model. brutus keeps the models whose weight is within a factor \(10^{-3}\) of the best (wt_thresh). If more than 50,000 remain (max_models), it keeps about 50,000 at random: model \(i\) is kept with probability \(p_i = \min(1, c\,w_i)\), proportional to this weight \(w_i\), and its posterior weight in step 4 is divided by \(p_i\). This keeps the evidence and the posterior draws unbiased.

3. Integrate the priors. For each kept model \(i\), brutus draws \(N_{\rm mc} = 50\) points from its Gaussian in \((\ln d, A_V, R_V)\), in antithetic pairs \((z, -z)\). Each point \(j\) is weighted by the priors that depend on position and by the exact parallax likelihood. The model’s posterior weight is the average:

\[ w_i \;\propto\; \mathcal{L}_{{\rm phot},i}\, \pi_{{\rm model},i} \; \frac{1}{N_{\rm mc}} \sum_{j=1}^{N_{\rm mc}} \pi_{\rm gal}(d_{ij}, \theta_i)\; \pi_{\rm dust}(A_{V,ij} \mid d_{ij})\; \mathcal{L}_\varpi(d_{ij})\; d_{ij}. \]

The factor \(d_{ij}\) is the Jacobian of sampling in \(\ln d\). Sampling in \(s\) instead would bias distances by a factor \(d^{-3}\).

The parallax is used three times. During step 1, models whose magnitude-space fit, with the parallax added at the best-fit \(s\), is more than a factor \(10^6\) worse than the best (logl_initthresh) keep that fit and are not refined. The averaged term in step 2 then decides which models are kept. Only the exact term here sets the weights.

4. Draw the posterior. brutus draws 250 models (Ndraws) in proportion to the posterior weight \(w_i\) of step 3, divided by \(p_i\) for the models kept by max_models, and for each one draws a point \((d, A_V, R_V)\) from its weighted Monte Carlo points. These are the saved posterior draws.

Why the parallax matters

Dust reddens a star as well as dimming it, so the colours measure the dust, but only as well as they measure the star’s temperature. A hotter star behind more dust has nearly the same colours as a cooler star behind less. On the main sequence the hotter star is also more luminous, so it lies farther away. Photometry alone therefore allows a nearer, less reddened star or a farther, more reddened one. A parallax limits the distance directly, and the posterior is where the two agree.

The distance-extinction degeneracyPhotometry alone allows a nearer, less reddened star or a farther, more reddened one. A parallax limits the distance, and the posterior is where both agree.distanceextinction A(V)photometry alone: nearer andless reddened, or farther and moreparallax: a rangeof distanceboth: the posterior

Without a parallax, the Galactic and dust priors set where along this degeneracy the posterior lies.

What brutus returns

For each star, the output file holds the posterior draws of distance, \(A_V\) and \(R_V\), the grid model of each draw (from which mass, age, temperature and the other properties follow), and fit diagnostics: the best \(\chi^2\), the number of bands, and the log-evidence. Understanding results describes each field.

An example: a sightline towards Orion

The tutorials include 207 stars in one small field towards Orion (\(\ell = 209.1°\), \(b = -19.9°\)), with Pan-STARRS and 2MASS photometry and Gaia parallaxes. Fitted with the default grid and priors, the stars that fit well trace the dust along the sightline: their extinction is low out to about 0.4 kpc and, with scatter, rises to 2 to 5 mag beyond 0.5 kpc, behind the Orion molecular cloud.

Distances and extinctions of stars towards OrionPosterior draws and medians of distance and extinction for 92 well-fitted stars within 2.5 kpc towards l = 209.1, b = -19.9. Extinction is low out to about 0.4 kpc and rises, with scatter, to 2 to 5 mag beyond 0.5 kpc, behind the Orion molecular cloud.distance (kpc)extinction A(V) (mag)0.511.522.50246mediandraws92 stars,p > 0.001,d < 2.5 kpc

Distance and extinction of the stars with p-value above \(10^{-3}\) (Understanding results) and median distance below 2.5 kpc, without a dust-map prior. Each star contributes 25 posterior draws and its median.

Fitting a dust profile to many such posteriors is how brutus maps dust in 3D; tutorial 7 shows it.

Assumptions

  • Single stars. The default grid has no unresolved binaries. A binary looks too bright, so its distance comes out too small.

  • Non-rotating, solar-scaled MIST models [Choi et al., 2016, Dotter, 2016]. Fast rotators and chemically peculiar stars are not modelled.

  • One extinction curve family, with \(R_V\) free within its prior.

  • Systematic differences between models and data remain at the level of a few hundredths of a magnitude per band. Photometric offsets describes the corrections and how to add their uncertainties to the errors.

For faint stars without a parallax, the result depends on the priors. Check how the result changes with them for your science case.