Part 6: Determining Main Sequence star properties from scratch#
Assuming a main sequence star (the rules change for white dwarf, giants, and supergiants), we will use only three pieces of empirical information to calculate the following stellar properties: distance, color, intrinsic brightness, temperature, stellar class, radius, and mass.
Observed information#
Magnitude through B-filter (Johnson): \(m_B=8.41\)
Magnitude through V-filter (Johnson): \(m_V=7.69\)
Parallax: \(\pi=0.0181242''\)
Calculate distance#
\(d=\frac{1}{\pi}=\frac{1}{0.0181242}=55.17\;pc\)
Calculate color index#
\(m_B-m_V=8.41-7.69=0.72\)
Calculate the absolute magnitude#
\(M_V=m_V-5\times \log \left(\frac{d}{10}\right)\)
\(M_V=4.81-5\times \log \left(\frac{55.17}{10}\right)\)
\(M_V=3.98\)
Calculate effective temperature#
\(T=4600\left(\frac{1}{0.92(B-V)+1.7}+\frac{1}{0.92(B-V)+0.62}\right)\)
\(T=4600\left(\frac{1}{0.92(0.72)+1.7}+\frac{1}{0.92(0.72)+0.62}\right)=5534\;K\)
Lookup spectral class#
| Spectral Class | Intrinsic Color | Temperature (K) | Prominent Absorption Lines |
|---|---|---|---|
| O | Blue | 30,000-60,000 | He+, O++, N++, Si++, He, H |
| B | Blue | 10,000-30,000 | He, H, O+, C+, N+, Si+ |
| A | Blue-white | 7500-10000 | H(strongest), Ca+, Mg+, Fe+ |
| F | White | 6000-7500 | H(weaker), Ca+, ionized metals |
| G | Yellow-white | 5000-6000 | H(weaker), Ca+, ionized & neutral metal |
| K | Orange | 3850-5000 | Ca+(strongest), neutral metals strong, H(weak) |
| M | Red | <3850 | Strong neutral atoms, TiO |
Based upon effective temperature, this star is probably G-class, yellow-white in color, with weaker hydrogen lines than F-class, and visible calcium ion lines in the spectrum.
Calculate luminosity#
Absolute luminosity
Recommended zero-point luminosity: \(L_0=3.0128\times 10^{28}\)
\(L=L_0\times 10^{\frac{-M}{2.512}}\)
\(L=3.0128\times 10^{28}\times 10^{\frac{-3.98}{2.512}}\)
\(L=7.84\times 10^{26}\;W\)
Luminosity compared to the Sun
Sun's Luminosity: \(L_s=3.828\times 10^{26}\)
Relative luminosity: \(L_r=\frac{L}{L_s}=\frac{7.84\times 10^{26}}{3.828\times 10^{26}}=2.05\)
This star is approximately 2 times the luminosity of the Sun
Lookup bolometric correction#
Bolometric correction for main sequence stars (Code, 1976)
An effective temperature of \(5534\;K\) and a \(B-V\) color index of \(0.72\) comes closest to \(5780\;K\) and \(0.63\), for a bolometric correction of \(BC=-0.07\).
Calculate radius#
Radius relative to the Sun
Use the following equation, with values for effective temperature, bolometric correction, and parallax to calculate the log of the radius ratio to the sun.
\(\log \frac{R}{R_s}=7.474-2\log T-0.2(BC)-0.2m-\log \pi\)
\(\log \frac{R}{R_s}=7.474-2\log 5534-0.2(-0.07)-0.2m-\log 0.0181242\)
\(\log \frac{R}{R_s}=0.21\)
\(\frac{R}{R_s}=10^{0.21}=1.61\)
This star is approximately 1.6 times the radius of the Sun.
Absolute radius
\(R_s=6.957\times 10^8\) meters
\(\frac{R}{R_s}=10^{0.21}\)
\(R=10^{0.21}\times R_s=1.12\times 10^{9}\) meters
Calculate mass#
Mass-Luminosity Relation: \(L=M^{3.5}\)
\(M=L^{\frac{2}{7}}\)
\(M=2.05^{\frac{2}{7}}=1.23\)
This is star is approximately 1.2 times the mass of the Sun.
These are the final results:
| property | value | obtained |
|---|---|---|
| B filter | 8.41 | measured |
| V filter | 7.69 | measured |
| parallax arcseconds | 0.018124 | measured |
| distance pc | 55.17 | calculated |
| color index | 0.72 | calculated |
| absolute magnitude | 3.98 | calculated |
| effective temperature | 5534 | calculated |
| spectral class | G | table |
| luminosity ratio to Sun | 2.05 | calculated |
| bolometric correction | -0.07 | table |
| radius ratio to Sun | 1.61 | calculated |
| mass ratio to Sun | 1.23 | calculated |
Searching in SIMBAD, here is a star which closely matches these parameters.
References#
Code, A. D., Bless, R. C., Davis, J., & Brown, R. H. (1976). Empirical effective temperatures and bolometric corrections for early-type stars. The Astrophysical Journal, 203, 417. https://doi.org/10.1086/154093