<p>A simplified argument is as follows.</p>
<p>The radius and speed of a Kepler ellipse with semi-major axis <span class="math-container">$a$</span> and eccentricity <span class="math-container">$e$</span> at apo- ('+') and perihelion ('<span class="math-container">$-$</span>') are,
<span class="math-container">$$
r_\pm = a(1\pm e),\qquad v_\pm = \sqrt{\frac{GM}{a}} \frac1{1\pm e}.
$$</span>
Hence
<span class="math-container">$$
a = \frac{r^2_+v^2_+}{GM},\qquad
e = \frac{GM}{r_+v^2_+} - 1,
$$</span>
and therefore
<span class="math-container">$$
r_- = a(1-e) = \frac{r^2_+v^2_+}{GM}\left(2-\frac{GM}{r_+v^2_+}\right)
= 2 \frac{r^2_+v^2_+}{GM} - r_+.
$$</span>
In particular, increasing <span class="math-container">$v_+$</span> decreases <span class="math-container">$e$</span> but increases <span class="math-container">$a$</span> and therefore also the perihelion distance <span class="math-container">$r_-$</span>. A passing star will essentially leave <span class="math-container">$r_+$</span> unchanged, but effectuate a small change in velocity. Since stars move much faster (relative to the Sun) than any Solar-system object, the general trend is to increase <span class="math-container">$v_+$</span>.</p>

CC BY-SA 4.0
2024-06-19T12:55:31.277Z
OK, that makes sense for a passing star, but the arguments for circularisation only involving interaction with solar system bodies still seem lacking.
John Dallman
CC BY-SA 4.0
2024-06-19T11:28:09.183Z
2024-06-19T11:28:09.183Z
Walter
<p>I've been reading about the hypothetical <a href="https://en.wikipedia.org/wiki/Planet_Nine#Origin" rel="nofollow noreferrer">Planet Nine</a>, but there are two passages in the Wikipedia article that puzzle me:</p>
<blockquote>
<p>Several possible origins for Planet Nine have been examined, including its ejection from the neighborhood of the known giant planets, capture from another star, and in situ formation. In their initial article, Batygin and Brown proposed that Planet Nine formed closer to the Sun and was ejected into a distant eccentric orbit following a close encounter with Jupiter or Saturn during the nebular epoch. Then, either the gravity of a nearby star or drag from the gaseous remnants of the Solar nebula reduced the eccentricity of its orbit. This process raised its perihelion, leaving it in a very wide but stable orbit beyond the influence of the other planets.</p>
</blockquote>
<p>It's easy to see how an encounter with another planet can make a planet's orbit more eccentric, raising the aphelion. I'm having more trouble understanding how the perihelion can get raised to circularise an orbit far from the Sun.</p>
<p>It's hard to say the gravity of another star couldn't do that, but it's an explanation that's very <em>convenient</em> for the theory.</p>
<p>However, I don't see how physical friction with the remaining gas of the solar nebula would do it. That will remove momentum and kinetic energy, but those need to be <em>added</em> to circularise an orbit.</p>
<blockquote>
<p>Dynamical friction from a massive belt of planetesimals also could have enabled Planet Nine's capture into a stable orbit. Recent models propose that a 60–130 ME disk of planetesimals could have formed as the gas was cleared from the outer parts of the proto-planetary disk. As Planet Nine passed through this disk its gravity would alter the paths of the individual objects in a way that reduced Planet Nine's velocity relative to it. This would lower the eccentricity of Planet Nine and stabilize its orbit.</p>
</blockquote>
<p>The same problem seems to apply to <a href="https://en.wikipedia.org/wiki/Dynamical_friction" rel="nofollow noreferrer">dynamical friction</a>. Again, it reduces momentum and kinetic energy.</p>
<p>What am I missing?</p>

CC BY-SA 4.0
2024-06-19T04:56:04.310Z
Removing momentum and energy near perihelion will circularize the orbit, just as adding energy near aphelion will (although the circular orbit will be smaller).  The interaction with the planetesimals would happen closer to the perihelion.  Although that wouldn't raise the perihelion. but perhaps that phrase was intended to be associated only with the interaction with the passing star.
antlersoft
<planetary-formation><orbital-migration>
How can a planet's perihelion be raised during migration?
John Dallman
# How can a planet's perihelion be raised during migration?

(Asked by: username_0 on 2024-06-18)

I've been reading about the hypothetical [Planet Nine](https://en.wikipedia.org/wiki/Planet_Nine#Origin), but there are two passages in the Wikipedia article that puzzle me:

> Several possible origins for Planet Nine have been examined, including its ejection from the neighborhood of the known giant planets, capture from another star, and in situ formation. In their initial article, Batygin and Brown proposed that Planet Nine formed closer to the Sun and was ejected into a distant eccentric orbit following a close encounter with Jupiter or Saturn during the nebular epoch. Then, either the gravity of a nearby star or drag from the gaseous remnants of the Solar nebula reduced the eccentricity of its orbit. This process raised its perihelion, leaving it in a very wide but stable orbit beyond the influence of the other planets.

It's easy to see how an encounter with another planet can make a planet's orbit more eccentric, raising the aphelion. I'm having more trouble understanding how the perihelion can get raised to circularise an orbit far from the Sun.

It's hard to say the gravity of another star couldn't do that, but it's an explanation that's very *convenient* for the theory.

However, I don't see how physical friction with the remaining gas of the solar nebula would do it. That will remove momentum and kinetic energy, but those need to be *added* to circularise an orbit.

> Dynamical friction from a massive belt of planetesimals also could have enabled Planet Nine's capture into a stable orbit. Recent models propose that a 60–130 ME disk of planetesimals could have formed as the gas was cleared from the outer parts of the proto-planetary disk. As Planet Nine passed through this disk its gravity would alter the paths of the individual objects in a way that reduced Planet Nine's velocity relative to it. This would lower the eccentricity of Planet Nine and stabilize its orbit.

The same problem seems to apply to [dynamical friction](https://en.wikipedia.org/wiki/Dynamical_friction). Again, it reduces momentum and kinetic energy.

What am I missing?

**Tags:** planetary-formation, orbital-migration
> **Comments:**
> **username_1**: Removing momentum and energy near perihelion will circularize the orbit, just as adding energy near aphelion will (although the circular orbit will be smaller).  The interaction with the planetesimals would happen closer to the perihelion.  Although that wouldn't raise the perihelion. but perhaps that phrase was intended to be associated only with the interaction with the passing star.  

---

## 1 answers

### Answer by username_2 on 2024-06-19. Score: 1

A simplified argument is as follows.

The radius and speed of a Kepler ellipse with semi-major axis $a$ and eccentricity $e$ at apo- ('+') and perihelion ('$-$') are,
$$
r\_\pm = a(1\pm e),\qquad v\_\pm = \sqrt{\frac{GM}{a}} \frac1{1\pm e}.
$$
Hence
$$
a = \frac{r^2\_+v^2\_+}{GM},\qquad
e = \frac{GM}{r\_+v^2\_+} - 1,
$$
and therefore
$$
r\_- = a(1-e) = \frac{r^2\_+v^2\_+}{GM}\left(2-\frac{GM}{r\_+v^2\_+}\right)
= 2 \frac{r^2\_+v^2\_+}{GM} - r\_+.
$$
In particular, increasing $v\_+$ decreases $e$ but increases $a$ and therefore also the perihelion distance $r\_-$. A passing star will essentially leave $r\_+$ unchanged, but effectuate a small change in velocity. Since stars move much faster (relative to the Sun) than any Solar-system object, the general trend is to increase $v\_+$.

> **Comments:**
> **username_0**: OK, that makes sense for a passing star, but the arguments for circularisation only involving interaction with solar system bodies still seem lacking.  

---

