Reproduced below is a set of expository notes I prepared in support of the August 13, 2016 installment of National Public Radio special series Joe's Big Idea, "Astronomers Are On A Celestial Treasure Hunt. The Prize? Planet Nine." The notes survey the mathematics and physics underpinning the search for Planet Nine. The reader is assumed to be familiar with basic algebra.

## 1. Ellipses

### 1.1. Circles

We begin by recalling that a circle of radius $$r$$ at center $$P$$ is the collection of all points of distance $$r$$ away from $$P$$.

### 1.2. Ellipses

We construct an ellipse, a flattened circle, as follows. Pick two points $$F_1$$ and $$F_2$$ on the plane, called foci.

Pick a third point $$x$$ and draw the lines $$\overline{xF_1}$$ and $$\overline{xF_2}$$.

We now draw a circular curve around the two foci in such a way that the sum of the length of the red line and the length of the blue line always remains constant.

### 1.3. Semi-major axis

Let us now draw a line through the foci, called the semi-major axis of the ellipse:

The sum of the length of the red line and the blue line equals the length of the semi-major axis.

### 1.4. Semi-minor axis

Let us find the center of the semi-major axis:

We now draw another line through the center, perpendicular to the semi-major axis:

The resulting line is called the semi-minor axis of the ellipse. Regardless of how we draw an ellipse, the semi-minor axis is never longer than the semi-major axis.

### 1.5. Extreme points

Let us now focus our attention on one of the two foci:

We draw the semi-major axis:

The closer endpoint $$a$$ is the point of least distance from $$F_1$$ on the ellipse. The further endpoint $$b$$ is the point of greatest distance from $$F_1$$ on the ellipse. These are important facts to consider, as we will model planetary motions with ellipses.

## 2. Newton's Universal Law of Gravitation

The mass of an object refers to the amount of matter in the object. Newton's second law of motion characterizes mass as a measurement of inertia, i.e., the resistance to acceleration:

$F = ma.$

Here $$F$$ refers to the total amount of force exerted on the object, $$m$$ the mass of the object, and $$a$$ the acceleration of the object. The identity implies that, given the same amount of force, more massive objects are less likely to change their states:

$a = \frac{F}{m}.$

Indeed, it is difficult to stop a moving object of great mass, or to move a stationary object of great mass.

Every object exerts an attractive force proportional to its mass, called gravitational force. Attraction is a concept that requires two objects — one "pulls the other — and so Newton's universal law of gravitation is stated for two objects of mass $$m_1$$ and $$m_2$$, separated by distance $$d$$:

$F = G \frac{m_1m_2}{d}.$

$$G$$ is a constant, called the universal constant of gravitation. We will not be doing numerical computations in this post, and so we may as well drop the constant:

$F \propto \frac{m_1m_2}{d^2}.$

The symbol $$\propto$$ denotes "is proportional to."

What does the law of universal gravitation say? $$F$$ refers to the amount of attractive force that exists between two objects. A larger numerator means a larger total quantity, and so the gravitational force $$F$$ is proportional to how massive the objects are: namely, $$m_1$$ and $$m_2$$.

The denominator $$d^2$$ implies that the gravitational force between two objects decreases as two objects move further away from each other. Precisely, gravitational force is inversely proportional to the square of the distance between the two objects. If the distance is increased twofold, the gravitational force will decrease by $$\frac{1}{4}$$. If the distance is increased tenfold, the gravitational force will decrease by $$\frac{1}{100}$$.

## 3. The Two-Body Problem

The study of celestial mechanics begins with a consideration of the two-body problem, where we assume that there are only two objects in the universe. For example, we might be interested in how the Sun and the Earth interact with each other. Or we could study how the Earth and the Moon interact with each other.

We are particularly interested in cases where one object is vastly more massive than the other, as in the examples cited above. In such cases, the lighter object revolves around the heavier object in an elliptical orbit, with the heavier object at one of the foci of the ellipsis, provided that the lighter object starts out with fast enough acceleration.

To understand how the gravitational force between the two objects results in an elliptical motion, we visualize the force by picturing an imaginary rope:

By Newton's law of inertia, the lighter object wishes to keep moving towards the direction it has been going.

But the "rope" pulls the lighter object towards the heavier object.

And so the trajectory of the lighter object is tilted towards the heavier object.

The heavier object pulls the lighter object closer and closer, and the trajectory changes accordingly.

By Newton's law of universal gravitation

$F_{\text{grav}} \propto \frac{m_{\text{light}}m_{\text{heavy}}}{d^2}.$

the gravitational force exerted on the lighter object increases as it approaches to the heavier object. Since we have assumed that there is no external force beyond the gravitational forces of the two objects, the total force $$F_{\text{total}}$$ on the lighter object increases as it approaches to the heavier object. Now, the mass $$m_{\text{light}}$$ of the lighter object remains constant, and so Newton's second law of motion

$F = ma$

implies that the acceleration $$a$$ increases as the lighter object approaches the heavier object.

Eventually, the lighter object reaches the semi-major axis of the ellipse.

Since we have assumed that the lighter object started out with a substantial forward-moving force, the increased acceleration now allows the lighter object to move away from the heavier object.

... and it orbits on and on and on.

We remark that the elliptical orbit model does not explain the movement of two objects of similar mass. For example, Pluto and Charon orbit each other, instead of one orbiting the other. Interested readers are encouraged to look up the concept of barycenter.

## 4. The $$n$$-Body Problem, Inverse Problems, and A Missing Planet

### 4.1. The $$n$$-Body Problem

unfortunately, there are more than two celestial objects in our solar system, and any physical model of the system must consider the gravitational interactions among multiple bodies. Unlike the two-body problem, there is no clean, exact solution for the $$n$$-body problem when $$n \geq 3$$. For this reason, astrophysicists prefer to think of an $$n$$-body problem as a collection of two-body problems. Specifically, we consider the interaction between the star and one of the planets and treat the gravitational pulls of the other celestial objects as perturbations — deviations — to the elliptical orbit of the planet.

The classic four-body problem in celestial mechanics is the Sun–Jupiter–Saturn–Uranus problem. Here Mercury, VEnus, Earth, Mars, and the asteroid belt are ignored, as their masses are negligible compared to the three gas giants or the sun.

Uranus was discovered by William Herschel in 1781 and its trajectory has been recorded regularly since. By 1847, Uranus had nearly completed one full orbit — now understood to be roughly 84 years — since its discovery.

### 4.2. Inverse Problems

The orbital path that the solution of the aforementioned four-body problem predicted was an inwardly-distorted ellipse, to account ofr the gravitational attractions of Jupiter and Saturn. Observational data, however, did not match this model, and it was suggested that an existence of a planet further away than Uranus, pulling the orbit outwards, could resolve the irregularities.

We pause here to note that finding the location of an unknown planet from known orbits — called an inverse problem — is a substantially more difficult problem than finding the orbit of a known planet in a known system — called a forward problem. The former is akin to figuring out the shape of the drum from the soundwaves the drum makes, whereas the latter is akin to figuring out the shapes of the soudnwaves made by a drum whose shape we know.

In other words, a forward problem is figuring out data points from a known graph and comparing them to observations, whereas an inverse problem is figuring out the graph that fits the available observational data.

Such problems, in general, cannot be solved exactly, as there are many models that can "fit" the same set of data. For example, if we have two points

a line could go through them

but so could many other curves

Therefore, in solving an inverse problem, it is necessary to put constraints on what the solution should look like. Such constraints often arise from our understanding of the physical world.

### 4.3. In Search of a Missing Planet

In case of the problem of finding a missing planet beyond Uranus, the recorded perturbations to the orbit of Uranus provided key constraints. Urbain Le Verrier wrote on June 1, 1846:

It would be natural to suppose that the new body is situated at twice the distance of Uranus from the Sun, even if the following considerations didn't make it almost certain. First, it is obvious that the sought-after planet cannot come too close to Uranus [since then its perturbations would have been very evident]. However, it is also difficult to place it as far off, say, as three times the distance of Uranus, for then we should have to give it an excessively large mass. But then its great distance both from Saturn and Uranus would mean that it would disturb each of these two planets in comparable degree, and it would not be possible to explain the irregularities of Uranus without at the same time introducing very sensible perturbations of Saturn, of which however there exist no trace.

We might add that since the orbits of Jupiter, Saturn, and Uranus all have a very small inclination to the ecliptic, it is reasonable to suppose, as a first approximation, that the same must apply to the sought-after planet.

(The ecliptic is the apparent path of the Sun on the celestial object.)

With the constraints, Le Verrier was able to determine the approximate location of a new planet, which was promptly verified at the Berlin Observatory by Johann Gottfried Galle and his student Heinrich Louis d'Arres on September 24, 1846. This planet is now known as Neptune.

## 5. Pluto and Planet Nine

The discovery of Pluto was a happy coincidence. The measured mass and orbit of Neptune could not account for the perturbations of Uranus's orbit completely, but then neither could Pluto.

While it is easy to speculate that there may be planets beyond Pluto, the approach used to find Neptune cannot be applied to the search of a trans-Plutonian planet. Pluto belongs to the Kuiper belt — a larger, more massive cousin of the asteroid belt between Mars and Jupiter — and it simply is not massive enough to be considered the sole representative of the belt.

What discrepancies, then, could we work with? the peculiar orbit of minor planet 90377 Sedna suggests that there must be a massive object in the outer reaches of the Solar system. Moreover, Konstantin Batygin and Michael E. Brown, in their 2016 paper "Evidence for a Distance Giant Planet in the Solar System," analyze the clustering of Kuiper Belt Objects (KBO). If we assume that there are no other planets in the solar system, then KBOs must cluster in accordance with the gravitational field of Neptune, the closest massive celestial object.

Computations show, however, that the clustering is "grossly inconsistent with the value predicted." Moreover, "the current mass of the Kuiper Belt is likely insufficient for self-gravity to play an appreciable role in its dynamical evolution."

Batygin and Brown set out to obtain suitable constraints for the inverse problem of finding the location of Planet Nine in their 2016 paper "Observational Constraints on the Orbit and Location of Planet Nine in the Outer Solar System." See Figure 10 of the paper for the predicted range of the location of Planet Nine.