<p>The current solar constant on the frost line is equal to:</p>
<p><span class="math-container">$$j_c=\frac{\sigma T_c^4 \cdot 4\pi r_c^2}{4\pi R_c^2}=\frac{\sigma T_c^4 r_c^2}{R_c^2}$$</span>
where <span class="math-container">$T_c$</span> is the current effective temperature, <span class="math-container">$r_c$</span> is the current radius of Sun and <span class="math-container">$R_c$</span> 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:
<span class="math-container">$$j=\frac{\sigma T^4 r^2}{R^2}$$</span>
We suppose, that the solar constant on the frost line needs to be the same in all eras. Therefore:
<span class="math-container">$$j_c=j$$</span>
<span class="math-container">$$\frac{\sigma T_c^4 r_c^2}{R_c^2}=\frac{\sigma T^4 r^2}{R^2}$$</span>
<span class="math-container">$$\frac{T_c^2r_c}{R_c}=\frac{T^2r}{R}$$</span></p>
<p>The data:</p>
<ul>
<li><span class="math-container">$T_c=5778\rm\,K$</span></li>
<li><span class="math-container">$r_c=7\cdot 10^8\rm\,m$</span></li>
<li><span class="math-container">$R_c=5.2\rm\,AU$</span></li>
<li><span class="math-container">$T\approx 3500\rm\,K$</span></li>
<li><span class="math-container">$r\approx 1\rm\,AU=1.5\cdot10^{11}\rm\,m$</span></li>
</ul>
<p>We can now express the radius of the frostline, <span class="math-container">$R$</span>
<span class="math-container">$$R=\left(\frac{T}{T_c}\right)^2 \frac{r}{r_c}R_c=\left(\frac{3500}{5778}\right)^2\frac{1.5\cdot10^{11}}{7\cdot10^8}5.2\rm\,AU\approx 400\rm\,AU$$</span></p>
<hr />
<p>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.</p>

CC BY-SA 4.0
2021-07-13T15:55:55.963Z
If the volatiles heated up and blown out passed the new frost line can recondense with each other or more likely dust, rocks, asteroids, and dwarf planets, why wouldn't they?
Brooks Nelson
CC BY-SA 4.0
2021-07-13T15:58:01.300Z
@BrooksNelson The gravitational force between them is still too minor.
User123
CC BY-SA 4.0
2021-07-13T20:12:48.750Z
Wouldn't the hydrogen bonds from the water molecules stick to each other and anything else polar?
Brooks Nelson
CC BY-SA 4.0
2021-07-13T20:50:00.957Z
@BrooksNelson The hydrogen bond isn't so strong in case of small amount of water molecules (the space is pretty much empty).
User123
CC BY-SA 4.0
2021-07-13T15:24:20.120Z
2021-07-13T15:24:20.120Z
User123
<p>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.</p>
<p>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.</p>

<the-sun><planetary-formation><kuiper-belt><red-giant><solar-system-evolution>
Where will the frost line be when the Sun becomes a red giant and what effect will it have on the solar system?
Brooks Nelson
# Where will the frost line be when the Sun becomes a red giant and what effect will it have on the solar system?

(Asked by: username_0 on 2021-07-13)

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.

**Tags:** the-sun, planetary-formation, kuiper-belt, red-giant, solar-system-evolution

---

## 1 answers

### Answer by username_1 on 2021-07-13. Score: 3

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, $R$
$$R=\left(\frac{T}{T\_c}\right)^2 \frac{r}{r\_c}R\_c=\left(\frac{3500}{5778}\right)^2\frac{1.5\cdot10^{11}}{7\cdot10^8}5.2\rm\,AU\approx 400\rm\,AU$$

---

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.

> **Comments:**
> **username_0**: If the volatiles heated up and blown out passed the new frost line can recondense with each other or more likely dust, rocks, asteroids, and dwarf planets, why wouldn't they?  
> **username_1**: @BrooksNelson The gravitational force between them is still too minor.  
> **username_0**: Wouldn't the hydrogen bonds from the water molecules stick to each other and anything else polar?  
> **username_1**: @BrooksNelson The hydrogen bond isn't so strong in case of small amount of water molecules (the space is pretty much empty).  

---

