How Bright Are Stars (Exactly)?

R. C. Bohlin (bohlin[at]

The brightness of a star means not just the total energy output, but also the amount of energy measured at the Earth as a function of wavelength in physical units of energy per cm2 per sec per unit wavelength, e.g., erg cm–2 s–1 Ångstrom–1, where an Ångstrom (Å) is 10–10 meter. This spectral energy distribution (SED) is also known as absolute spectral flux by astronomers, or as irradiance by physicists. The flux of a star is commonly measured with telescopic spectrometers relative to a standard star with a known SED. The variable transmission of the Earth's atmosphere hinders accurate flux measurements from the ground, which means that there are few reliable absolute flux measures in the literature. Thus, HST flux standards use a model atmosphere to establish the relative flux vs. wavelength, which is then normalized to the absolute flux level of the few reliable absolute measures, such as the stars Vega at a visible wavelength and Sirius at infrared wavelengths. Accurate standard star SEDs are important for measuring relative fluxes of redshifted supernovae Ⅰa's and, thus, in determining the nature of the dark energy that is driving the observed accelerating cosmic expansion.

The most reliable model atmosphere calculations should be for stellar atmospheres consisting of the simplest atom, hydrogen; and some hot white dwarf (WD) stars do have pure hydrogen atmospheres. Historically, the uncertainty of these WD model flux distributions was set by the amount of discrepancy between different modeling efforts; but recently, new pure hydrogen model atmosphere grids computed independently by Ivan Hubeny and by Thomas Rauch agree much better (Bohlin, Hubeny, & Rauch 2020). Thus, with formal error bars that decrease to less than 1% from near-ultraviolet to mid-infrared wavelengths, these new models define primary standards with more exact SEDs. The new WD pure hydrogen grids are available in the MAST Archive.

Another advantage of WD stars is that the brightest are nearby with little attenuation by interstellar dust, which is the case for the three HST primary standards G191B2B, GD153, and GD71 with known and almost negligible interstellar reddening and effective temperature in the 30000–60000 K range.

ratio of TAMP standards by wavelength
Figure 1: Change in the SED for each of the three primary flux standards and the average change due to changes in the reference models. Because the absorption lines are masked in the flux calibration process, the sharp spikes at the line centers have no effect on the results. Previously, the Rauch TMAP2012 code was used to compute the model SEDs.

Cooler WDs may have more precise models in the infrared but are less certain at far-UV wavelengths. Figure 1 shows the change in these primary SEDs with respect to the previous calibration of Bohlin, Gordon, & Tremblay (2014). In addition to these wavelength-dependent changes, there is also a gray flux increase of 0.87%, corresponding to the new normalization value of 3.44 x 10–9 erg cm–2 s-1 Å–1 at the monochromatic wavelength of 5556 Å for Vega.

The revised flux distributions of the three primary HST standard stars means that the flux calibration of all the HST instruments must be updated. Following a recalibration of the STIS low dispersion modes and application of the new flux calibration to the STIS observations of the three primary standards, the residuals of the calibrated data with respect to their new models illustrate the internal consistency of the chosen model parameters, as shown in Figure 2. The measured relative flux distributions of the three primaries is nearly the same as the relative flux among the chosen models, which suggests that the models are correct to 0.5%, typically, and to 1% at the Balmer confluence at 3650–4100 Å. Hopefully, these small discrepancies average out in any calibration that is based on the model SEDs in comparison to an equal weight of observations for all three stars. However, for calibrations in the wavelength range of STIS from 1150–10000 Å, the use of the STIS SEDs in preference to the model SEDs would improve precision in the Balmer confluence region, where small modeling errors are apparent. The models have no Poisson noise but that noise source is minimal in the final CALSPEC1 SEDs, which consist of averages of 13–36 independent STIS observations.

Both the model and the observed reference SEDs are available from the CALSPEC database, along with many secondary standards with fluxes from STIS, WFC3, and NICMOS spectrophotometry, where those instrumental calibrations are based on the three new primary WD model SEDs. WFC3 grism data extend the STIS wavelength coverage to 1.7 micron, while the NICMOS spectra extend to 2.5 micron. See the related article in this newsletter by Calamida et al.

STIS consistency by wavelength
Figure 2: Residuals for the new set of primary standard WDs. STIS SEDs from the five low-dispersion observing modes in the 52 x 2 slit are combined to make the numerator flux distribution, while the denominators are the new model SEDs calculations for GD153, GD71, and G191B2B. The average and rms values written on each panel are computed in the range delineated by vertical dotted lines at 1750 Å and 8000 Å.

While the internal consistency of the three primary WD models suggests that rms uncertainties on the new flux scale are ~0.5% from 0.4–30 micron, comparison with independent external flux distribution measurements adds confidence. Recent measures of the stellar 109 Vir SED along with the older Vega SED from STIS are compared with the absolute fluxes from Tüg et al. (1977) that have a claimed rms precision of 1%. Figure 3 shows the ratio of the Tüg et al. (1977) results to the CALSPEC alpha_lyr_stis_010.fits and 109vir stis 002.fits SEDs. The ratios and rms are roughly consistent with the Tüg et al. (1977) uncertainty of 1% longward of 4000 Å, where most of the points are within 2σ of unity, and only the one point for Vega at 5890 Å deviates by as much as 4σ. The good agreement of the independent absolute flux measures in Figure 3 bound the errors in the HST/CALSPEC flux scale to <2% in the wavelength range of Figure 3 and are consistent with the smaller internal uncertainties.


Bohlin, R. C., Gordon, K. D., & Tremblay, P.-E. 2014, PASP, 126, 711          

Bohlin, R. C., Hubeny, I., & Rauch, T. 2020, AJ, 160, 21                                         

Tüg, H., White, N. M., & Lockwood, G. W. 1977, A&A, 61, 679