202004240535 Homework 1 (Q4)

What is the effective temperature of a neutron star with radius $10^6\,\mathrm{cm}$ which is detected to emit an energy flux $F=9\times 10^{-15}\,\mathrm{erg\, s^{-1}\, cm^{-2}}$ ? The distance to this neutron star is $1\,\mathrm{kpc}$ .

Setup.

Recall that the effective temperature $T_{\textrm{eff}}$ of a blackbody defines the frequency-integrated intensity of radiation at the source:

$\sigma_{\textrm{SB}}T^4_{\textrm{eff}}=F=\displaystyle{\int}I_\nu\cos\theta\,\mathrm{d}\nu\,\mathrm{d}\Omega$

where $\sigma_{\textrm{SB}}=5.67\times 10^{-5}\,\mathrm{erg\cdot cm^{-2}\, K^{-4}}$ is the Stefan-Boltzmann constant.

Solution. (guesstimate)

The luminosity $L$ of a source is related to the energy flux $F$ detected at some distance $d$ away from the source by

$F=\displaystyle{\frac{L}{4\pi d^2}}$ ,

from which one can infer the luminosity of the neutron star in question is

$\begin{aligned} L & =9\times 10^{-15}\times 4\pi (3.085677581\times 10^{21})^2 \\ & = 1.076845664\times 10^{30}\enspace \mathrm{erg\, s^{-1}} \end{aligned}$

where $d=1\,\mathrm{kpc}=3.085677581\times 10^{21}\,\mathrm{cm}$ is given.

Since $L=4\pi R^2\sigma_{\textrm{SB}}T_{\textrm{eff}}^4$ ,

$\begin{aligned} T^4_{\textrm{eff}} & =L/4\pi R^2\sigma_{\textrm{SB}} \\ T_{\textrm{eff}} & = \bigg(\frac{L}{4\pi R^2\sigma_{\textrm{SB}}}\bigg)^{1/4} \\ & = \bigg(\frac{1.076845664\times 10^{30}}{4\pi \times (10^6)^2\times 5.67\times 10^{-5}}\bigg)^{1/4} \\ T_{\textrm{eff}} & = 197169.6816\enspace \mathrm{K} \end{aligned}$

M	T	W	T	F	S	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30

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