Sunday, April 5, 2015

Worksheet 12.2, Problem 3: Habitable Zone and Stellar Mass

In this problem we'll figure out how the habitable zone distance, \(a_{HZ}\) depends on stellar mass. Recall the average mass-luminosity relation that we derived earlier, as well as the mass-radius relation for stars in the main sequence.  
(a) Express \(a_{HZ}\) in terms of stellar properties as a scaling relationship.
(b) Replace the stellar parameters with their dependence on stellar mass.
(c) If the Sun were half as massive and the earth had the same equilibrium temperature, how many days would our year be?



First let's recall some equations from previous problems:
\[L \sim M^4\]
\[M \sim R_*\]
\[L \sim R_*^2 T_{eff}^4\]
\[T_p^2 = \frac{T_{eff}^2R_*}{2a}\]
\[L \sim M_*^4\]
Where \(L\) is luminosity, \(M\) is mass, \(T_p\) is planetary temperature, \(T_{eff}\), \(a\) is semi major axis, and \(R_*\) is radius.

(a) Express \(a_{HZ}\) in terms of stellar properties as a scaling relationship.
To express \(a_{HZ}\) in terms of stellar properties we should solve the fourth equation for \(a\).
\[T_p^2 = \frac{T_{eff}^2R_*}{2a}\]
\[a=\frac{T_{eff}^2R_*}{2T_p^2}\]
Since \(2T_p^2\) is a constant, in the scale equation we can omit it.
\[\boxed{a \sim T_{eff}^2R_*}\]

(b) Replace the stellar parameters with their dependence on stellar mass.
By taking the square root of both sides of the third equation we get:
\[\sqrt{L} \sim R_* T_{eff}^2\]
We can substitute this into our previous answer:
\[a \sim T_{eff}^2R_*\]
\[a \sim \sqrt{L}\]
We can solve for L using equation 5.
\[a \sim \sqrt{L}\]
\[\boxed{a \sim M^2}\]

(c) If the Sun were half as massive and the earth had the same equilibrium temperature, how many days would our year be?
This calls for the Third Law of Planetary Motion:
\[P^2 \propto \frac{4\pi^2 a^3}{GM_*}\]
Where \(P \) is orbital period and \(G\) is Newton's gravity constant.
We don't need any constants, so we can remove those.
\[P^2 \propto \frac{a^3}{M_*}\]
We can solve this in terms of \(M_*\) from the previous part.
\[P^2 \propto \frac{(M_*^2)^3}{M_*}\]
\[P^2 \propto M_*^5\]

Now the problem postulates that \(M_* \rightarrow \frac{M_*}{2}\)
Let's set up a proportion:
\[\frac{P_0^2}{M_*^5}=\frac{P^2}{(\frac{M_*}{2})^5}\]
We can cancel a \(M_*^5\) from both sides to get:
\[P_0^2=32P^2\]
Solving this for our new period gives us:
\[P=\frac{P_0}{\sqrt{32}}\]
We know that our original period was 365.25 days, so:
\[P=\frac{365.25\text{ days}}{\sqrt{32}}\]
\[\boxed{P\approx 65\text{ days}}\]

I solved this problem in collaboration with G. Grell and S. Morrison.

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