Blog #28 - Habitable Zone pt. 2

Habitable Zone pt.2

2. In this problem we’ll figure out how the habitable zone distance, Ahz, depends on stellar mass. Recall the average mass-luminosity relation we derived earlier, as well as the mass-radius relation for stars on the main sequence. If you don’t recall, this is a good time to practice something that will very likely show up on the final!

(a) Express aHZ in terms of stellar properties as a scaling relationship, using squiggles instead of equal signs and ditching constants.

Simplifying the answer from part one, we get the relation \[T_p \approx C*L_s^{\frac{1}{4}}*a_{hz}^{\frac{1}{2}} \]Rearranging for a we get \[a_{hz}^{\frac{1}{2}} \approx C*L_s^{\frac{1}{4}}*T_p^{-1}\]Which gives us \[a_hz \approx C*L_s^{\frac{1}{2}}*T_p^{-2} \]

(b) Replace the stellar parameters with their dependence on stellar mass, such that aHZ ∼ Mα. Find α.

From an earlier worksheet we know the relation that \[L_s \approx M_s^4 \] This relation can help turn the semi major axis into a number based solely on the physical makeup of the star and planet. From part (a) we have \[a_hz \approx \frac{L^{\frac{1}{2}}}{T_{hz}^{2}} \approx \frac{M_s^{2}}{T_{hz}^{2}}\]

(c) If the Sun were half as massive and the Earth had the same equilibrium temperature, how many days would our year contain?

Converting this to a relationship for the orbital period, we have \[P^2 \approx \frac{a^3}{M_s}\] Substituting in our term for \(a_{hz}\) we get \[P^2 \approx \frac{(\frac{M_s^2}{T_{hz}^2})^3}{M_s} \]Using the information that \(T_{hz}\) is the same and \(M = .5 M_s \) we get \[P^2 \approx \frac{(\frac{\frac{1}{2}M_s^2}{T_{hz}^2})^3}{\frac{1}{2}M_s} \approx \frac{\frac{1}{64}M_s}{\frac{1}{2} M_s} \rightarrow P \approx \sqrt{\frac{1}{32}}*P_E \]Giving us a final answer of \[P \approx \sqrt{\frac{1}{32}}*365 \space days = 64.52 \space days \]