<p>If we have two masses <span class="math-container">$M$</span> and <span class="math-container">$m$</span>, which are at distances <span class="math-container">$r_1$</span> and <span class="math-container">$r_2$</span> respectively from their barycentre, then
<span class="math-container">$$Mr_1 = mr_2$$</span></p>
<p>Let <span class="math-container">$d$</span> equal the total distance between the two bodies. That is, <span class="math-container">$d = r_1 + r_2$</span>. Then
<span class="math-container">$$r_1 = \frac{md}{M+m}$$</span>
where <span class="math-container">$r_1$</span> is the distance of the barycentre from the body with mass <span class="math-container">$M$</span>.</p>
<p>The paper linked in the comments suggests a figure of 500 au for the distance between the Sun and the alleged Planet 9. Let <span class="math-container">$M$</span> be the mass of the Sun, and <span class="math-container">$m$</span> be the mass of Planet 9. Using units of Earth masses, we have <span class="math-container">$M=333000$</span> and <span class="math-container">$m=5$</span>.</p>
<p>Plugging those numbers into our equation, we get
<span class="math-container">$$r_1 = \frac{500×5}{333005} \approx 0.0075074$$</span>
for the distance (in au) of the barycentre from the centre of the Sun.</p>
<p>Now, the <a href="https://en.wikipedia.org/wiki/Solar_radius" rel="nofollow noreferrer">solar radius</a> (the nominal radius of its photosphere) is <span class="math-container">$0.00465047$</span> au. So the barycentre is well outside the Sun. It's roughly <span class="math-container">$1.61$</span> solar radii from the centre of the Sun, or <span class="math-container">$\approx 427400$</span> km above the surface of the photosphere.</p>
<hr />
<p>As James K points out, to accurately calculate the orbit of a body in the Solar System we need to include the masses of all the other bodies. Of course, that's impractical. The best ephemerides are produced by JPL, and their <a href="https://en.m.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris" rel="nofollow noreferrer">Development Ephemeris series</a> currently uses the masses and locations of all major Solar System bodies down to the 340 most significant asteroids.</p>
<p>As you can see <a href="https://astronomy.stackexchange.com/a/28036/16685">here</a>, the motion of the Solar System barycentre (SSB) is not simple. And its true location is <em>not</em> precisely known, mostly because we don't know the details of the mass distribution in the far outer reaches of the Solar System. In fact, recent updates to the JPL Horizons system have modified their value for the SSB.</p>
<p>From <a href="https://ssd.jpl.nasa.gov/?horizons_news" rel="nofollow noreferrer">Horizons System News</a>:</p>
<blockquote>
<h3>April 12, 2021</h3>
<p>The current catalog of 1.1 million asteroid and comet solutions are being refit for dynamical consistency with DE441 perturbations. The new solutions will filter into the database over the next days. Due to the addition of KBO mass in DE440/441, the SSB has shifted about 100 km</p>
</blockquote>

CC BY-SA 4.0
2021-06-11T11:35:12.923Z
Interesting. Your linked answer too.
John
CC BY-SA 4.0
2021-06-11T13:00:43.350Z
I'm having a fundamental problem understanding the concepts here, I'm somehow missing something. The solar system barycenter is moving... *with respect to what?* Some group of distant pulsars?
uhoh
CC BY-SA 4.0
2021-06-11T13:16:37.560Z
@uhoh I think you meant quasars, not pulsars. See the [International Celestial Reference System](https://en.wikipedia.org/wiki/International_Celestial_Reference_System). The Solar System as a whole has peculiar velocity relative to the Milky Way, and of course the Milky Way has its own motion relative to the Local Group, etc. But all of that isn't very relevant in the context of Solar System dynamics. For Solar System ephemerides, it's nice to use a frame with the SSB as the origin. But as I noted above, we don't really know its true location, so we use a "nominal" SSB.
PM 2Ring
CC BY-SA 4.0
2021-06-11T22:46:09.503Z
@PM2Ring everything moves with respect to everything but when we talk about "the motion of the SSB" we must include "with respect to X" within the same sentence otherwise it doesn't have a clear meaning.
uhoh
CC BY-SA 4.0
2021-06-11T22:51:11.120Z
@PM2Ring I'm thinking of the recent NANOGrav results: [Modeling the uncertainties of solar-system ephemerides for robust gravitational-wave searches with pulsar timing arrays](https://arxiv.org/abs/2001.00595) and [...Within the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), uncertainties on the solar system barycenter are emerging as an important element of the NANOGrav noise budget....](https://arxiv.org/abs/1801.02898)
uhoh
CC BY-SA 4.0
2021-06-11T22:52:02.800Z
@uhoh In the graphics in my linked answer, (both the images I copied from Wikipedia & my 3D plot), the motion of the SSB is shown relative to the centre of the Sun.
PM 2Ring
CC BY-SA 4.0
2021-06-11T22:53:42.207Z
@PM2Ring okay I'll take a look, thanks!  **update:** Okay well I'd call that *the motion of the Sun* with respect to the solar system barycenter rather than vice-versa. Either way it's motion of X with respect to Y. And I think the whole dance of all solar system objects is being nailed down with pulsars, not quasars. But I am no expert. I'll keep reading...
uhoh
CC BY-SA 4.0
2021-06-11T23:01:14.130Z
The [IERS](https://www.iers.org/IERS/EN/DataProducts/ICRF/icrf.html) has a lot of info about the ICRS. I must confess that I've only looked at a few of their (& NASA's) documents on this topic. They have some great info, but there's also a lot of boring bureaucratic stuff. ;)
PM 2Ring
CC BY-SA 4.0
2021-06-11T23:04:43.020Z
To create my plot I query Horizons for the 3D position & velocity vectors of the Sun in the SSB frame, and simply reverse those vectors so I can use the Sun as the origin of the plot.
PM 2Ring
CC BY-SA 4.0
2021-06-11T11:13:40.090Z
2022-11-19T12:51:43.687Z
2022-11-19T12:51:43.687Z
PM 2Ring
<p>The proposal is that the apparent planet 9 is, in fact, a primordial black-hole, with mass comparable to a planet and a diameter measured in centimeters. Such an object would be almost undetectable.</p>
<p>Its orbit would be identical to an equivalent sized planet.</p>
<p>In a two body system, obeying Newtonian gravity, both bodies will follow elliptical orbits, with the barycentre at their focus. The barycentre will be at rest, so it is reasonable to make a coordinate frame in which the barycenter is at (0,0) and the two objects orbit around it.</p>
<p>In the case of a three (or more) body system, the orbit won't be elliptical. So the notion of a point around which the body orbits becomes less clear. It will be affected by the gravity of all the other solar system bodies, and it's acceleration vector (in an inertial frame) will point towards the centre of mass of the other solar system bodies, but this point is would not be fixed.</p>

CC BY-SA 4.0
2021-06-09T18:08:30.243Z
Fun aside: The first paper I linked to above includes an actual size illustration of the hypothetical 5M⊕ black hole (the picture is a 4.5cm radius black circle).
Alex Hajnal
CC BY-SA 4.0
2021-06-09T18:04:23.243Z
2021-06-09T18:04:23.243Z
James K
<p>In 2016, an additional object rotating our Sun has been suggested. Assuming it has 5 Earth masses, would the planet or black hole proposed by <a href="https://arxiv.org/abs/1601.05438" rel="noreferrer">Brown and Batigyn</a> have a barycenter within the Sun or outside, or might it orbit the Sun-Jupiter barycenter, like <a href="https://en.wikipedia.org/wiki/90377_Sedna" rel="noreferrer">Sedna</a>?</p>

CC BY-SA 4.0
2021-06-09T17:14:27.490Z
Do you have a link to this (rather daring) hypothesis?
planetmaker
CC BY-SA 4.0
2021-06-09T17:25:48.030Z
@planetmaker Dr. Becky [mentioned this in a video](https://www.youtube.com/watch?v=AplFPtHY09E&t=830s) in 2019.  [Here's the paper](https://arxiv.org/pdf/1909.11090.pdf) she linked to (*What if Planet 9 is a Primordial Black Hole?*) which cites this paper:  K. Batygin, and M. E. Brown, *Evidence for a distant giant planet in the solar system*, AJ 151.2 (2016): 22. [1601.05438](https://arxiv.org/pdf/1601.05438.pdf)
Alex Hajnal
CC BY-SA 4.0
2021-06-09T17:53:39.960Z
My question asks where the barycenter of the hypothetical object would be, not if it is a planet or black hole (which we don't know, not even if the object exists).
John
CC BY-SA 4.0
2021-06-09T17:56:33.017Z
@John, planetmaker was asking for a reference to papers positing the hypothesis.
Alex Hajnal
CC BY-SA 4.0
2021-06-11T14:31:24.810Z
@John Thanks for the clarifying edit!
B--rian
CC BY-SA 4.0
2021-06-11T18:30:20.120Z
Related: https://astronomy.stackexchange.com/q/36894/34443
ksousa
<solar-system><orbital-elements><hypothetical>
If the hypothesized planet behind the Kuiper belt existed, would it have a barycenter outside the Sun?
John
# If the hypothesized planet behind the Kuiper belt existed, would it have a barycenter outside the Sun?

(Asked by: username_0 on 2021-06-09)

In 2016, an additional object rotating our Sun has been suggested. Assuming it has 5 Earth masses, would the planet or black hole proposed by [Brown and Batigyn](https://arxiv.org/abs/1601.05438) have a barycenter within the Sun or outside, or might it orbit the Sun-Jupiter barycenter, like [Sedna](https://en.wikipedia.org/wiki/90377_Sedna)?

**Tags:** solar-system, orbital-elements, hypothetical

> **Comments:**
> **username_1**: Do you have a link to this (rather daring) hypothesis?  
> **username_2**: @username_1 Dr. Becky [mentioned this in a video](https://www.youtube.com/watch?v=AplFPtHY09E&t=830s) in 2019.  [Here's the paper](https://arxiv.org/pdf/1909.11090.pdf) she linked to (*What if Planet 9 is a Primordial Black Hole?*) which cites this paper:  K. Batygin, and M. E. Brown, *Evidence for a distant giant planet in the solar system*, AJ 151.2 (2016): 22. [1601.05438](https://arxiv.org/pdf/1601.05438.pdf)  
> **username_0**: My question asks where the barycenter of the hypothetical object would be, not if it is a planet or black hole (which we don't know, not even if the object exists).  
> **username_2**: @username_0, username_1 was asking for a reference to papers positing the hypothesis.  
> **username_3**: @username_0 Thanks for the clarifying edit!  
> **username_4**: Related: https://astronomy.stackexchange.com/q/36894/34443  

---

## 2 answers

### Answer by username_7 on 2021-06-09. Score: 4 | Accepted answer

The proposal is that the apparent planet 9 is, in fact, a primordial black-hole, with mass comparable to a planet and a diameter measured in centimeters. Such an object would be almost undetectable.

Its orbit would be identical to an equivalent sized planet.

In a two body system, obeying Newtonian gravity, both bodies will follow elliptical orbits, with the barycentre at their focus. The barycentre will be at rest, so it is reasonable to make a coordinate frame in which the barycenter is at (0,0) and the two objects orbit around it.

In the case of a three (or more) body system, the orbit won't be elliptical. So the notion of a point around which the body orbits becomes less clear. It will be affected by the gravity of all the other solar system bodies, and it's acceleration vector (in an inertial frame) will point towards the centre of mass of the other solar system bodies, but this point is would not be fixed.

> **Comments:**
> **username_2**: Fun aside: The first paper I linked to above includes an actual size illustration of the hypothetical 5M⊕ black hole (the picture is a 4.5cm radius black circle).  

---

### Answer by username_5 on 2021-06-11. Score: 7

If we have two masses $M$ and $m$, which are at distances $r\_1$ and $r\_2$ respectively from their barycentre, then
$$Mr\_1 = mr\_2$$

Let $d$ equal the total distance between the two bodies. That is, $d = r\_1 + r\_2$. Then
$$r\_1 = \frac{md}{M+m}$$
where $r\_1$ is the distance of the barycentre from the body with mass $M$.

The paper linked in the comments suggests a figure of 500 au for the distance between the Sun and the alleged Planet 9. Let $M$ be the mass of the Sun, and $m$ be the mass of Planet 9. Using units of Earth masses, we have $M=333000$ and $m=5$.

Plugging those numbers into our equation, we get
$$r\_1 = \frac{500×5}{333005} \approx 0.0075074$$
for the distance (in au) of the barycentre from the centre of the Sun.

Now, the [solar radius](https://en.wikipedia.org/wiki/Solar_radius) (the nominal radius of its photosphere) is $0.00465047$ au. So the barycentre is well outside the Sun. It's roughly $1.61$ solar radii from the centre of the Sun, or $\approx 427400$ km above the surface of the photosphere.

---

As username_7 points out, to accurately calculate the orbit of a body in the Solar System we need to include the masses of all the other bodies. Of course, that's impractical. The best ephemerides are produced by JPL, and their [Development Ephemeris series](https://en.m.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris) currently uses the masses and locations of all major Solar System bodies down to the 340 most significant asteroids.

As you can see [here](https://astronomy.stackexchange.com/a/28036/16685), the motion of the Solar System barycentre (SSB) is not simple. And its true location is *not* precisely known, mostly because we don't know the details of the mass distribution in the far outer reaches of the Solar System. In fact, recent updates to the JPL Horizons system have modified their value for the SSB.

From [Horizons System News](https://ssd.jpl.nasa.gov/?horizons_news):

> ### April 12, 2021
>
> The current catalog of 1.1 million asteroid and comet solutions are being refit for dynamical consistency with DE441 perturbations. The new solutions will filter into the database over the next days. Due to the addition of KBO mass in DE440/441, the SSB has shifted about 100 km

> **Comments:**
> **username_0**: Interesting. Your linked answer too.  
> **username_6**: I'm having a fundamental problem understanding the concepts here, I'm somehow missing something. The solar system barycenter is moving... *with respect to what?* Some group of distant pulsars?  
> **username_5**: @username_6 I think you meant quasars, not pulsars. See the [International Celestial Reference System](https://en.wikipedia.org/wiki/International_Celestial_Reference_System). The Solar System as a whole has peculiar velocity relative to the Milky Way, and of course the Milky Way has its own motion relative to the Local Group, etc. But all of that isn't very relevant in the context of Solar System dynamics. For Solar System ephemerides, it's nice to use a frame with the SSB as the origin. But as I noted above, we don't really know its true location, so we use a "nominal" SSB.  
> **username_6**: @username_5 everything moves with respect to everything but when we talk about "the motion of the SSB" we must include "with respect to X" within the same sentence otherwise it doesn't have a clear meaning.  
> **username_6**: @username_5 I'm thinking of the recent NANOGrav results: [Modeling the uncertainties of solar-system ephemerides for robust gravitational-wave searches with pulsar timing arrays](https://arxiv.org/abs/2001.00595) and [...Within the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), uncertainties on the solar system barycenter are emerging as an important element of the NANOGrav noise budget....](https://arxiv.org/abs/1801.02898)  
> **username_5**: @username_6 In the graphics in my linked answer, (both the images I copied from Wikipedia & my 3D plot), the motion of the SSB is shown relative to the centre of the Sun.  
> **username_6**: @username_5 okay I'll take a look, thanks!  **update:** Okay well I'd call that *the motion of the Sun* with respect to the solar system barycenter rather than vice-versa. Either way it's motion of X with respect to Y. And I think the whole dance of all solar system objects is being nailed down with pulsars, not quasars. But I am no expert. I'll keep reading...  
> **username_5**: The [IERS](https://www.iers.org/IERS/EN/DataProducts/ICRF/icrf.html) has a lot of info about the ICRS. I must confess that I've only looked at a few of their (& NASA's) documents on this topic. They have some great info, but there's also a lot of boring bureaucratic stuff. ;)  
> **username_5**: To create my plot I query Horizons for the 3D position & velocity vectors of the Sun in the SSB frame, and simply reverse those vectors so I can use the Sun as the origin of the plot.  

---

