Question

Explain how we estimate the effective surface temperature of the Sun. A main
sequence star has mass 2 × 10^31 kg and radius 3 × 10^9 m. Obtain an estimate of the
average temperature throughout the star.

Answer 1

Answer on Question #69698, Physics / Astronomy | Astrophysics

Explain how we estimate the effective surface temperature of the Sun.

Answer:

According to Wien's Law,

$$\lambda_ {m a x} = \frac {2 9 0 0 0 0 0}{T}$$

$\lambda_{max} =$ wavelength of maximum intensity, $b =$ Wien's constant $= 2,900,000$ nm.K, $T =$ temperature in Kelvin

The color at a wavelength of $500\mathrm{nm}$ that corresponds to in the visible-light spectrum is green. The human eye is most sensitive for peaks at this wavelength. This wavelength is the peak length of thermal emission from the Sun.

$$T = \frac {2 9 0 0 0 0 0}{\lambda_ {m a x}}$$

Then,

$$T = \frac {2 9 0 0 0 0 0}{5 0 0} = 5 8 0 0 K$$

A main sequence star has mass $2 \times 10^{\wedge}31$ kg and radius $3 \times 10^{\wedge}9$ m. Obtain an estimate of the average temperature throughout the star.

Answer:

The average temperature $< T >$ will lie between $T_{c}$ and $T_{s}$ .

We can use the virial theorem to find $< T>$

$$2 K E _ {t o t} + P E _ {t o t} = 0$$ $$2 \left(\frac {3}{2} N k < T >\right) - \frac {3}{5} G \frac {m _ {H} M ^ {2}}{R} = 0$$

Where, $\mathrm{N} = \mathrm{M} / \mu \mathrm{m}_{\mathrm{H}}$ , let $\mu = 1$

$$< T > = \frac {1}{5} G \frac {m _ {H} M}{k R}$$

$G = 6.7 \cdot 10^{-11} \, \text{N} \, \text{m}^2 / \text{kg}^2$ (constant of gravitation), $m_H = 1.7 \cdot 10^{-27} \, \text{kg}$ (mass of hydrogen atom), $M = 2 \cdot 10^{31} \, \text{kg}$ (mass of the Star), $k = 1.4 \cdot 10^{-23} \, \text{J} / \text{K}$ (Boltzmann constant), $R = 3 \cdot 10^9 \, \text{m}$ (radius of the Star)

Then,

$$< T > = \frac {1}{5} \times 2 \cdot 1 0 ^ {- 1 1} \times \frac {1 . 7 \cdot 1 0 ^ {- 2 7} \times 2 \cdot 1 0 ^ {3 1}}{1 . 4 \cdot 1 0 ^ {- 2 3} \times 3 \cdot 1 0 ^ {9}} = 3. 2 4 \cdot 1 0 ^ {6} K$$