# 天文学入门|PHYS1160 Introduction to Astronomy代写

1. One definition of the habitable zone, is that region of space around a star, where the average temperature of a planetary body would be between $0^{\circ} \mathrm{C}$ and $100^{\circ} \mathrm{C}$.
(a) Calculate the habitable zone for the Solar System. To do so, equate the incoming flux from the Sun with the outgoing flux from the planet’s black-body radiation (use the Stefan-Boltzmann law for this). In considering the incident solar radiation, assume an albedo (i.e. the fraction of incident light that is directly reflected) of $0.31$, as for the Earth .

Solution:
The incident radiation from the Sun can be calculated from its luminosity (which in turn could have been derived from its black-body flux as given by the Stefan-Boltzmann law; integrated over its surface):
$$F_{\mathrm{in}}=\frac{L_{\mathrm{a}}}{4 \pi D^{2}}$$
with $D$ the distance between the Sun and the Earth (i.e. $1 \mathrm{AU}$ ). The energy absorbed by the Earth , then becomes:
$$E_{\text {in }}=F_{\text {in }} \pi R_{\text {Earth }}^{2}(1-A)=\frac{L_{\square} R_{\text {Earth }}^{2}(1-A)}{4 D^{2}}$$
where $A=0.31$ is the Earth’s albedo,i.e. the fraction of the incoming flux that is not absorbed. The energy lost in black-body radiation, is given by:
$$E_{\text {out }}=4 \pi R_{\text {Earth }}^{2} \sigma T_{\text {Earth }}^{4}$$
with $\sigma T_{\text {Earth }}^{4}$ the blackbody flux as defined by the Stefan-Boltzmann law.
Equating these two, we get:
$$4 \pi R_{\text {Earth }}^{2} \sigma T_{\text {Earth }}^{4}=\frac{L_{\odot} R_{\text {Earth }}^{2}(1-A)}{4 D^{2}}$$
or, after re-w riting:
For $T=273 \mathrm{~K}$, this becomes :
$$D_{\max }=1.29 \times 10^{11} \mathrm{~m}=0.86 \mathrm{AU}$$
and for $T=373 \mathrm{~K}$, we get:
$$D_{\text {min }}=6.94 \times 10^{10} \mathrm{~m}=0.46 \mathrm{AU} .$$
This is obviously a suspicious result, because the Earth (which must be in the habitable zone), is not included. The reason this is the case, is because we ignored many details, the most important one being the green house effect, which heats planets and therefore moves the habitable zone further out.

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