Bolometric Correction

Astronomy 218 What color is that Star? Colored Stars When you look at the stars in the night sky, another Meissa noticeable feature, for a Betelgeuse few stars, is their color. Bellatrix To the naked eye, only a few bright stars have colors Mintaka discernibly different than Alnilam Alnitak white, because your color vision is less sensitive that your monochrome vision. Hatsya Rigel The Sun’s yellow color is Saiph an obvious example. Betelgeuse is another example. Spectra The perceived color or a star (or any other object) is a product of its spectrum. A star has a flux that depends on wavelength (or frequency), Fλ. This is defined such that Fλ dλ is the star’s flux in the range of wavelengths from λ to λ+dλ. The total, or bolometric, flux is Blackbody Spectrum An opaque object emits a characteristic, temperature dependent, spectrum. Wien’s Law 3 2.898 10− λ = × The Planck function max T explained the observed shape of the Blackbody. There are two useful limits to consider. For low energy photons, For high energy photons, E << kT, we have the E >> kT, we have the Rayleigh-Jeans Limit. Wien Limit. Blackbody Flux The flux of energy emitted per unit area, per unit time in a interval of wavelength, can be calculated by integrating the intensity over the solid angle. The result, -1 -1 Fλ = πIλ has units of s m . The total flux is calculated by integrating over wavelength -8 -1 -2 -4 where σSB = 5.67×10 J s m K Total Luminosity is Flux × area of emitting surface. 2 4 L = 4πR σSB T Bolometric Magnitude From the bolometric flux, it is natural to calculate a bolometric apparent magnitude, mbol = Cbol − 2.5 log Fbol For visual apparent magnitude, C was chosen based on Vega having a nearly zero and very stable apparent magnitude. However, it is hard to measure fluxes across all wavelengths, thus the uncertainties in the bolometric flux are relatively large. For this reason, the Sun was chosen to be the “zero” of bolometric magnitude, with the Sun’s absolute bolometric magnitude defined as Mbol,☉ = 4.74. Astronomical Colors The challenge to measuring a bolometric flux is that all detectors have a limited range of wavelength over which they can detect light, so different detectors yield different measurements. For example, your eye detects light from 400-700 nm. Astronomical colors are measured using filters to select specific wavelength bands (photometry). One standard set of photometric filters is the UBV(RIJ…) set originated by Harold Johnson & William Wilson Morgan in 1953. Filtered Fluxes Any detector or filter has a spectral sensitivity S(λ) that quantifies how easily light of a given frequency is detected. The flux measured is a convolution of the emitted flux and the detector’s sensitivity. Where X could be a broadband filter like photographic (pg), U, B, V, R, I, … or a narrowband filter. Each filter can be approximately described by a bandwidth, Δλ, and an effective wavelength Filter Colors From these filtered fluxes, one can calculate apparent magnitudes. Particularly common are the standard UBVRI apparent magnitudes. mU = CU − 2.5 log FU mB = CB − 2.5 log FB mV = CV − 2.5 log FV ︙ In keeping with astronomical tradition, CU, CV, CB … are chosen such that Vega has mU = mB = mV … ~ 0 (+0.03). The difference between adjacent apparent magnitudes is a color index. The most common example is B−V ≡ mB − mV Other common color indices include U−B and V−R. Color Temperature For blackbody radiation, observations in 2 wavebands are enough to define a color temperature, because the Planck function is asymmetric. By convention, mB = mV ~ 0 for Vega. Thus B−V = 0 for Vega, or any star with the same temperature ~ 10,000K. Hotter stars will have B−V < 0 and cooler stars have B−V > 0. Because stars aren’t exactly blackbodies, it is convenient to use an empirical relation, which is valid from 4000 K to 11000 K. Why Photometry? Given the wealth of detail revealed in a star’s spectrum, why do we not use spectroscopy exclusively? Photometry is faster. A few exposures reveals the apparent magnitudes and colors for an entire field of stars. Bolometric Correction Since bolometric luminosities are so difficult to measure, it would be convenient if there was a correction that would convert, for example, mV to mbol. We define the bolometric correction such that BC = mbol − mV = Mbol − MV For a blackbody, BC is straightforward to calculate, but for realistic stars, stellar models provide a better basis. For the Sun Mbol ≡ 4.74 and MV = 4.83, thus BC = −0.09. In general, BC < 0 for stars with temperatures < 6700 K. At 6700 K, the peak emission falls squarely in the V band, thus MV ~ Mbol. BC is positive for T > 6700 K. Effective Temperature For stars, the color indices and bolometric correction allow determination of the temperature and luminosity. More generally, for an object radiating as a black body, one can use Wien’s law to find the temperature. For an blackbody with a known radius, we can also calculate a temperature from 2 4 L = 4πR σSB T Even for a non-blackbody we can calculate But Teff ≠ T if the object is not a blackbody. Estimating Stellar Sizes Observations over the late 19th and early 20th centuries revealed stars of similar color but with wide ranges in absolute magnitude. For example, both Proxima Centauri and Betelgeuse have effective temperatures near 3000 K, but differ widely in absolute magnitude, 15.5 and −6.1, respectively. Variations of 20 magnitudes in brightness among objects radiating with similar spectra led quickly to the need for a wide variety of stellar sizes. 2 4 L = 4πR σSB T Reconciling the luminosity of Proxima Centauri to that of Betelgeuse requires a factor of nearly difference in 10,000 in radius. Betelgeuse In 1920, the angular size of Betelgeuse was measured with an interferometer, providing the first direct measurement of the size of a star besides the Sun. The development of aperture mask or speckle interferometry, allowed diffraction-limited imaging of bright sources from the ground. Speckle imaging of Betelgeuse, beginning in 1980s and 1990s, refined the measured size. These images also revealed surface features thought to result from convective cells, like the Sun but on a larger scale. The Size of Betelgeuse In 1995, Betelgeuse was directly imaged in the UV by Hubble Space Telescope, to that date the only star to be imaged, with an angular diameter α = 0.13” = 3.5 × 10−5° = 6.1 × 10−7 rad. For a distance d, the stellar radius R is given by For d = 131 pc, R = dα/2 = 8.2 AU. = 1800 R☉. However, recent parallax 1 AU measurements vary from 5 mas to 10 mas (100-200 pc). Wavelength Dependence In comparison to the UV measured size of 130 mas, optical and near IR observations reveal angular diameters of 55 mas and 45 mas, respectively. This is thought to be the result of an extended chromosphere. Moreover, the angular diameter was observed to decline from 56 mas in 1993 to 47 mas in 2008, a contraction of 0.9 AU. Stellar Sizes Radio arrays, like the Very Large Array (VLA), and more recently optical interferometers, like the Very Large Telescope (VLT), have been used to measure the sizes of nearly a thousand nearby stars. For example, the stars of the α Centauri system have radii of 1.23 R☉ (α Cen A), 0.87 R☉ (α Cen B) & 0.14 R☉ (Proxima Centauri). Such measurements provide calibration for the bolometric corrections. Dwarves and Giants For the vast majority of stars that cannot be resolved, size must be calculated from the luminosity and temperature. Stellar radii vary widely •Dwarf stars have radii equal to, or less than, the Sun’s. •Giant stars have radii 10 - 100 times the Sun’s. •Supergiant star radii are > 100 R☉. •There is no correlation between color and size. Betelgeuse, 1180 R☉ For Example Epsilon Eridani is a neighbor of the Sun, located 3.22 pc away, near Sirius in the sky. Its apparent magnitudes are mB = 4.61 & mV = 3.73. B-V = 0.88, making ε Eri redder than Vega. Using ≈ 5000 K The corresponding BC is −0.40, mbol = 3.73 − 0.40 = 3.33. The absolute magnitude is Mbol = mbol − 5 log(d/10 pc) = 5.80 The Luminosity is = 0.38 The Radius is = 0.8 R☉ A wide variety of Stars Combining measurements of brightness and distance with temperature and radius reveals a menagerie of stars in our solar neighborhood. Star Luminosity Temp. Radius Sirius B 0.025 4.7 0.007 = 0.8 R⨁ Proxima Cen 0.0017 0.52 0.15 = 3.9 RNeptune Barnard’s Star 0.0045 0.56 0.2 = 2.0 RJupiter Sun 1 1 1 Sirius A 23 2.1 1.9 Vega 55 1.6 2.8 Arcturus 160 0.78 21 Rigel 63,000 1.9 70 Betelgeuse 120,000 0.55 >900 > 2.0 aMars Alpha Centauri Though invisible to the naked eye, our near neighbor Alpha Centauri Proxima is in fact a double star. Centauri: First noted as a binary in 1689, 0.002 L☉ the pair of stars α Cen A & B 0.14 R☉ circle each other with a period of 79.91 years. A: 1.52 L☉ 1.23 R☉ An angular separation of 2 – 22” reveals an eccentric orbit with a separation varying from 11.2 – B: 0.50 L☉ 0.87 R☉ 35.6 AU.

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