Where will the frost line be when the Sun becomes a red giant and what effect will it have on the solar system?

Question

I understand the frost line is currently about 5.2 AU and earlier in the solar systems formation was 2.7 AU. But when the Sun becomes a red giant the frost line should move outward. I understand the habitable zone will even include Jupiter and Saturn, which if we extrapolate means that the frost line would be further out than that, possibly near the orbit Uranus or Neptune.

The other part of the question is what effect this will have. If there is a planet near the frost line, I would think it would potentially increase in mass as the recondensed volatiles collect at the new frost line and falls into the planets orbit. (Or is flung into or out of that orbit.) If it is not near a planet, I would think matter and volatiles would collect there into a planet. Whatever Kuiper belt object is near enough may start to gain mass and eventually coalesce into a planet as it clears the orbit around it.

Answer 1

The current solar constant on the frost line is equal to:

$$j_c=\frac{\sigma T_c^4 \cdot 4\pi r_c^2}{4\pi R_c^2}=\frac{\sigma T_c^4 r_c^2}{R_c^2}$$ where $T_c$ is the current effective temperature, $r_c$ is the current radius of Sun and $R_c$ is the current distance of frost line. We write the equation for solar constant on the frost line when the Sun is a red giant: $$j=\frac{\sigma T^4 r^2}{R^2}$$ We suppose, that the solar constant on the frost line needs to be the same in all eras. Therefore: $$j_c=j$$ $$\frac{\sigma T_c^4 r_c^2}{R_c^2}=\frac{\sigma T^4 r^2}{R^2}$$ $$\frac{T_c^2r_c}{R_c}=\frac{T^2r}{R}$$

The data:

• $T_c=5778\rm\,K$
• $r_c=7\cdot 10^8\rm\,m$
• $R_c=5.2\rm\,AU$
• $T\approx 3500\rm\,K$
• $r\approx 1\rm\,AU=1.5\cdot10^{11}\rm\,m$

We can now express the radius of the frostline, $R$ $$R=\left(\frac{T}{T_c}\right)^2 \frac{r}{r_c}R_c=\left(\frac{3500}{5778}\right)^2\frac{1.5\cdot10^{11}}{7\cdot10^8}5.2\rm\,AU\approx 400\rm\,AU$$

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But no, the Kuiper belt objects won't grow up. That's because for objects on the new frost line, there won't be any recondensation.