If the hypothesized planet behind the Kuiper belt existed, would it have a barycenter outside the Sun?

Question

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 have a barycenter within the Sun or outside, or might it orbit the Sun-Jupiter barycenter, like Sedna?

Answer 1

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 (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.

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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 Development Ephemeris series currently uses the masses and locations of all major Solar System bodies down to the 340 most significant asteroids.

As you can see here, 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:

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

Answer 2

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.