Stellar models

brutus turns the intrinsic parameters of a star into its stellar properties with the MIST v1.2 models [Choi et al., 2016, Dotter, 2016], and its properties into photometry with a neural network. This page describes both steps. Model grids describes the pre-computed grids built from them.

Evolutionary tracks and EEP

A MIST track follows one star of fixed initial mass and composition through its life. brutus indexes each track by the equivalent evolutionary point, not by age. Points with the same EEP are at the same phase on every track, so tracks of different masses can be interpolated at fixed EEP. Age follows from (initial mass, EEP, [Fe/H]). Along a track, age changes by very different amounts per step: a star spends most of its life on the main sequence, so the prior includes the time per unit of EEP (How brutus works).

The primary EEPs mark the same physical transition on every track [Dotter, 2016]:

EEP	Phase
1	start of the pre-main sequence
202	zero-age main sequence (ZAMS)
353	intermediate-age main sequence: central hydrogen fraction 0.3
454	terminal-age main sequence (TAMS): central hydrogen exhausted
605	tip of the red giant branch
631	zero-age core helium burning
707	end of core helium burning
808	start of the thermally pulsing asymptotic giant branch

The default grid spans EEP 202 to 808: from the ZAMS to the thermally pulsing AGB.

Tracks of the default grid at solar metallicity. The dots mark the primary EEPs from the ZAMS to the TP-AGB; on the 1 M☉ track they are labelled.

Tracks and isochrones

brutus reads the MIST models in two forms:

	Evolutionary tracks	Isochrones
Fixed	initial mass, composition	age, composition
Varies along it	EEP	initial mass
Parameter classes	EEPTracks	Isochrone
Photometry classes	StarEvolTrack, GridGenerator	StellarPop
Used for	individual stars and the model grids	star clusters (Star clusters)
Data file	fetch_tracks()	fetch_isos()

EEPTracks.get_predictions takes [mini, eep, feh, afe] and returns the columns named in tracks.predictions: by default log age, log luminosity, log effective temperature, log surface gravity, the surface [Fe/H] and [α/Fe], and the age weight agewt.

Empirical corrections

By default (apply_corr=True), brutus corrects the MIST temperatures and radii of main-sequence stars below 1 M☉, where the models are systematically offset from observations:

\[ \Delta \log_{10} T_{\rm eff} = f\, \log_{10}\!\left(1 + 0.09\,\Delta M\right), \qquad \Delta \log_{10} R = f\, \log_{10}\!\left(1 - 0.09\,\Delta M\right), \qquad f = \frac{e^{0.5\,{\rm [Fe/H]}}}{1 + e^{({\rm EEP} - 454)/30}}, \]

with \(\Delta M = M_{\rm init} / M_\odot - 1\). The factor \(f\) falls from about \(e^{0.5\,{\rm [Fe/H]}}\) on the main sequence to 0 after the turnoff (EEP 454). corr_params sets the four coefficients (0.09, −0.09, 30, 0.5).

From stellar parameters to photometry

A neural network (FastNNPredictor, file nn_c3k.h5 from fetch_nns()) predicts the bolometric correction \({\rm BC}_b\) in each band \(b\) from the effective temperature, surface gravity, surface [Fe/H], [α/Fe], \(A_V\) and \(R_V\). The extinction is part of the network’s input, so the reddened magnitude comes out directly:

\[ m_b = -2.5 \log_{10} \frac{L}{L_\odot} + 4.74 - {\rm BC}_b\left(T_{\rm eff}, \log g, {\rm [Fe/H]}_{\rm surf}, {\rm [\alpha/Fe]}, A_V, R_V\right) + \mu(d), \]

where 4.74 is the bolometric magnitude of the Sun and \(\mu(d) = 5 \log_{10}(d / 10\,{\rm pc})\) is the distance modulus. The network returns NaN outside the range it was trained on: \(T_{\rm eff}\) from 2,500 to 50,000 K, \(\log g\) from −1 to 5.5, [Fe/H] from −4 to +0.5, [α/Fe] from −0.2 to +0.6, \(A_V\) from 0 to 6 and \(R_V\) from 2 to 5.

A model grid does not call the network during a fit. It stores each model’s magnitudes at 1 kpc and two reddening vectors fitted to the network’s output (Model grids).

Binaries

StarEvolTrack.get_seds and StellarPop.get_seds can add an unresolved companion, whose flux is added to the primary’s:

• StarEvolTrack.get_seds(smf=...) sets the secondary mass fraction of one star.

• StellarPop.get_seds(binary_fraction=...) sets the same mass ratio for every star of a population. Despite its name, it is a mass ratio, not a fraction of stars that are binaries.

Companions are added only to primaries with EEP up to eep_binary_max (480 by default). The default grid has no binaries (Model grids).

Available filters

The network and the default grid cover 49 bands. brutus.data.filters holds each survey’s list (for example filters.gaia or filters.tmass), and filters.FILTERS holds all of them.

Survey	List	Bands
Gaia	gaia	Gaia_G_MAW, Gaia_BP_MAWf, Gaia_RP_MAW
SDSS	sdss	SDSS_u, SDSS_g, SDSS_r, SDSS_i, SDSS_z
Pan-STARRS	ps	PS_g, PS_r, PS_i, PS_z, PS_y, PS_w, PS_open
DECam	decam	DECam_u, DECam_g, DECam_r, DECam_i, DECam_z, DECam_Y
Johnson-Cousins	bessell	Bessell_U, Bessell_B, Bessell_V, Bessell_R, Bessell_I
2MASS	tmass	2MASS_J, 2MASS_H, 2MASS_Ks
VISTA	vista	VISTA_Z, VISTA_Y, VISTA_J, VISTA_H, VISTA_Ks
UKIDSS	ukidss	UKIDSS_Z, UKIDSS_Y, UKIDSS_J, UKIDSS_H, UKIDSS_K
WISE	wise	WISE_W1, WISE_W2, WISE_W3, WISE_W4
Tycho	tycho	Tycho_B, Tycho_V
Hipparcos	hipp	Hipparcos_Hp
Kepler	kepler	Kepler_D51, Kepler_Kp
TESS	tess	TESS

Limitations

• Rotation: the default models and grid do not rotate. fetch_isos(iso="MIST_1.2_vvcrit0.4") gives isochrones with rotation for population work.

• Composition: the default grid is solar-scaled ([α/Fe] = 0).

• Stellar types: the models do not include white dwarfs, brown dwarfs or binary evolution (mass transfer, mergers).

• Model systematics: the empirical corrections reduce, but do not remove, the offsets of low-mass stars. Photometric offsets describes the per-band corrections that remain.