Can mathematics predict a planet’s motion?

We can look at a planet today. But how can we say where it will be a month from now, before anyone has seen that sky? Drawing an orbit is not enough: it shows a path without telling us when the planet will reach each part of it.

Mathematics can predict a planet’s position when we supply a model of its motion and an initial state. New observations test how reliable that prediction is. The initial state includes position and velocity at a specified instant; velocity has both a magnitude and a direction. JPL: how orbits and ephemerides are determined.

A path needs a clock and a cause of motion

Draw a closed curve and put a dot on it, and many possibilities remain. The body could move in either direction, faster or slower. Even the same starting point does not fix the future. We need to know how the body is moving now and what rule changes that motion.

In a physical model, that rule is a law of motion under forces. A planet continues moving through inertia while the star’s gravity changes its velocity. In a circular orbit, gravity continually turns the velocity toward a new tangent direction. With the appropriate initial speed, the distance from the star stays constant. This is a special case, not the outcome of every possible launch. OpenStax: the mechanics of circular orbits.

The causal chain is therefore forces and initial state → changes in motion → positions at later times. In a general problem, a computer calculates these changes step by step. Some simplified cases have a mathematical solution that links time directly to position. We will use one to learn how to read a prediction.

Build one imaginary orbit with explicit assumptions

Imagine a planet orbiting a star much more massive than itself. We neglect the star’s own motion and other bodies’ effects. The orbit is a circle of radius 1 au, with one revolution taking 360 days. These are deliberately chosen teaching numbers for an imaginary system, not Earth’s year or a forecast for the Solar System. One astronomical unit, au, is approximately 150 million kilometres. JPL: the definition of au.

Radius, period and the star’s gravity must be compatible with circular motion: around a given real star, we cannot choose any radius and period independently. Here we specify a possible idealized system so that reading the graph can be separated from the more involved force calculation.

Choose fixed axes in the orbital plane. The star is the origin; positive $x$ points right and positive $y$ points up. We look perpendicular to the plane. At $t=0$ , the planet is to the star’s right and moves counterclockwise. The period is the time for a complete revolution, returning to the same position with the same direction of motion.

In uniform circular motion, a quarter of the period covers a quarter of the circle. After 90 days the planet is at the top, after 180 at the left, after 270 at the bottom, and after 360 at the right again. We now have specific positions at specific times, rather than just the shape of a path.

A coordinate graph is a different picture from an orbit

In the first plot, both axes describe space: where the planet is right or left, above or below the star. In the second, the horizontal axis is time and the vertical axis is just coordinate $x$ . Point B marks the same instant in both plots, but their axes mean different things.

One orbit, two different plots

Teaching model: R = 1 au, T = 360 days. This is not Earth’s orbit.

Looking at the orbital plane

Coordinate x over time

The star is the origin; the axes stay fixed. We look perpendicular to the plane. Motion follows A → B → C → D → E.

The same times in both plots — calculated model values
Point	t (days)	Angle	x (au)	y (au)
A	0	0°	+1	0
B	90	90°	0	+1
C	180	180°	−1	0
D	270	270°	0	−1
E	360	360°	+1	0

x = 0 does not put the planet at the star. B and D have different y coordinates. Dots and star are enlarged; the points are not observations.

On the graph, $x$ falls from +1 au to −1 au and then rises again. This does not mean the planet moves back and forth along a straight line. Only its projection onto the $x$ axis does that while the body itself circles the star. Its distance from the star remains 1 au throughout; a coordinate can be negative, a distance cannot.

B and D make the distinction especially clear. Both have $x=0$ , but at B the planet is above the star and at D it is below. One coordinate cannot recover the full position. We also need $y$ , or an angle and a radius. Moreover, $x$ changes at a varying rate even though speed around the circle is constant: near the right and left edges, motion is mainly along $y$ .

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How time becomes two coordinates

Let $R$ be the radius, $T$ the period, and $\theta$ the angle measured from positive $x$ . With our starting point and direction of motion:

\theta(t)=360^\circ\frac{t}{T}.

Time $t$ and period $T$ must use the same units. Their ratio is the fraction of a revolution. Cosine and sine turn the angle into the radius’s projections onto the two axes:

\begin{aligned} x(t)&=R\cos\theta(t),\\ y(t)&=R\sin\theta(t). \end{aligned}

This is a parameterization: a way to specify coordinates through a shared parameter, here time. The cosine curve follows from that geometry; it was not fitted by eye to a set of dots. Writing sine and cosine does not, by itself, explain which force sustains the orbit.

For example, after 45 days:

\begin{aligned} \frac{t}{T}&=\frac{45}{360}=\frac18,\\ \theta&=45^\circ,\\ x&\approx0.707\,\mathrm{au},\\ y&\approx0.707\,\mathrm{au}. \end{aligned}

The planet is at the upper right, not 0.707 au from the star: the two coordinates together give a radius of 1 au. At $t=0$ , the formulas return $(R,0)$ ; at $t=T$ , the same position after one revolution. Software sin/cos functions usually expect angles in radians: replace $360^\circ$ with $2\pi$ in the first formula.

The plotted prediction still needs a comparison with the sky

Every dot in our figure was calculated using the same rule. Agreement between dots and curve checks consistency between the drawing and the calculation. It does not yet show that any real planet moves that way.

To test a prediction, first fix the model and its parameters, then compare it with new measurements. For a sky position seen from Earth, calculate the corresponding direction from the observer. We cannot simply equate it with a coordinate in a star-centred system. A precise comparison also accounts for the time light takes to reach the observer. JPL Horizons: coordinates and observer tables.

Assess the discrepancy together with measurement uncertainty. It may point to an inaccurately determined initial state, an omitted effect, or an observational problem. New data help refine the solution; one agreement does not make a model infallible at every future instant.

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How fitting differs from testing

Suppose we selected a period and a starting position so the curve would pass close to known measurements. That is parameter fitting. It is necessary because a real body’s initial state is never known with unlimited precision. But those measurements have already influenced the choice of curve; agreement with them is not an independent test.

It helps to ask whether the prediction agrees with an observation that was not used in the fit. Does the model remain useful at another time? Reproducing a past interval well does not guarantee a good continuation far beyond it.

JPL separately publishes approximate planetary-position formulas with fitted orbital elements and rates of change. They have stated time intervals and error estimates; the shape of the formulas does not justify extending them beyond those intervals. JPL: approximate positions of the planets.

What real ephemerides add

Real planets do not go around the Sun uniformly on identical circles. In the two-body approximation, orbits are ellipses; a planet moves faster nearer the Sun. NASA: Kepler’s laws. Bodies’ mutual gravity further changes their motion, so it does not repeat perfectly. JPL: the limits of a fixed orbit.

Ephemerides give calculated positions and other properties of motion at specified times. For example, JPL made DE440 and DE441 by solving equations of motion numerically and fitting the result to ground-based and space-based observations. This is a dated example of a real calculation, not a name for our circular model. Park and colleagues, 2021: the original DE440/DE441 paper.

For a request to a system such as Horizons, specify the body, time, centre or observer, coordinate system and units. “Where will Mars be?” is incomplete without these settings. An ephemeris is a model result limited by data and computational accuracy; it is not a photograph of the future.

Back to our teaching diagram: after 450 days, coordinate $x$ is zero again. Has the planet returned to its starting point? No. 450 days is a full revolution plus a quarter. The planet is at B: $x=0$ , $y=+1$ au. It returned to A after 360 days. We need both time and the meaning of the coordinates, not merely the height of a dot on the graph.

Motion can therefore be predicted: physical laws and an initial state give future positions, while new observations test accuracy. A graph makes that link visible when we understand its axes and the model’s limits. Look again at the contents of the Solar System: each additional body in that picture can become part of a more accurate calculation.