Two bodies, collisions and slow changes

Builds on: Sections 3.1, 3.2 and 3.5
Reading time: 35 min
Exercises: 5
Olympiad problems: 3

Jupiter does not go round a fixed Sun. The Sun and Jupiter both go round their common centre of mass, which lies 742 300 km from the centre of the Sun, just outside its surface. Once every 11.86 years the Sun moves round a small circle at 12.5 m/s. An astronomer on another star could detect Jupiter from that wobble alone, and this is how many planets round other stars were found (Chapter 9). In this section we treat both bodies as moving, and then look at what happens to an orbit when something changes, suddenly or slowly.

3.7.1Two bodies and their centre of mass

Before you readMake a guess first

Earth and the Moon go round their common centre of mass. Where is it?

Each of two bodies pulls the other with the same force, so neither can stay still. Their centre of mass, or barycentre, the point that divides the line between them in the inverse ratio of their masses, moves steadily or stays at rest, and both bodies go round it. Each moves on an ellipse of the same shape and eccentricity, with the centre of mass at a focus, and the two are always on opposite sides of it. The heavier body has the smaller ellipse and moves more slowly.

Figure 3.6Two bodies round their centre of mass

Mass of 2 over mass of 1Eccentricity

e

Watch

B lies at
Speeds

The figure shows two bodies, 1 and 2, moving round their common centre of mass B, with body 2 the lighter of the two. Both follow ellipses of the same shape with B at a focus, and the dashed line between them always passes through B. The sizes of the two ellipses, and the speeds of the bodies, are in the inverse ratio of the masses. The figure can be set to , , whose masses are in the ratio 8 to 1 on a circular orbit, or , where Earth barely moves.

The separation of the two bodies follows an ellipse of the same shape at full size, with semi-major axis $a = a_{1} + a_{2}$ . The pull on each body depends on the total mass, which is why Kepler's third law has $M + m$ in it ((3.2)).

ResultTwo bodies round their centre of mass(3.16)

\begin{aligned} m_{1} a_{1} & = m_{2} a_{2}, a = a_{1} + a_{2} \\ \frac{v_{1}}{v_{2}} & = \frac{a_{1}}{a_{2}} = \frac{m_{2}}{m_{1}} \\ T^{2} & = \frac{4 π^{2} a^{3}}{G (m_{1} + m_{2})} \end{aligned}

$m_{1}, m_{2}$	the masses of the two bodies
$a_{1}, a_{2}$	the semi-major axes of their ellipses round the centre of mass
$a$	the semi-major axis of their separation
$v_{1}, v_{2}$	their speeds at any one moment
$T$	the period, the same for both

a_{1} = a m_{2} / (m_{1} + m_{2})

. Measuring the period and the separation gives the total mass, and measuring how far or how fast each body moves round the centre of mass splits it into the two masses. Chapter 8 uses this to weigh binary stars.

Quick checkWhere the centre of mass lies

Two stars of 2 and 1 solar masses go round each other 3 au apart. How far is the heavier star from their centre of mass?

Example 3.12The Sun's wobble

Jupiter has a mass of 1/1047.6 of the Sun's, orbits at 5.203 au and moves at 13.06 km/s. How far is the Sun's centre from the centre of mass of the two, and how fast does the Sun move round it? The Sun's radius is 695 700 km.

Solution

By (3.16), the Sun's distance from the centre of mass is

a_{⊙} = \frac{5.203 \times 1.496 \times 10^{8} km}{1048.6} = 742 300 km,

which is 1.07 times the Sun's radius. The speeds are in the same ratio as the distances, so the Sun moves at $13.06 km/s / 1047.6 = 12.5$ m/s.

AnswerThe centre of mass is 742 300 km from the Sun's centre, just outside its surface, and the Sun moves round it at 12.5 m/s. The other planets add smaller wobbles of their own, so the centre of mass of the whole Solar System wanders, sometimes inside the Sun and sometimes outside it.

3.7.2Collisions and sudden kicks

An impact or an explosion is over in a moment, far shorter than an orbit. During that moment gravity has no time to act, so the total momentum of the bodies that collide stays the same. After it, each body follows a new orbit through the point of the collision, with its new velocity, exactly as after a rocket burn (Section 3.5). When the change of speed is small, (3.12) gives the change of the orbit at once. The DART spacecraft, which struck the small moon of an asteroid in 2022 to test whether an asteroid could be pushed off course, is the problem below.

3.7.3Slow changes

When the central mass changes slowly, over many orbits, the orbit has time to adjust, and it stays nearly circular. Gravity still pulls straight towards the centre, so the angular momentum stays the same. For a circular orbit $v = \sqrt{G M / r}$ , so the angular momentum of each kilogram is $r v = \sqrt{G M r}$ , and keeping it fixed keeps the product $M r$ fixed.

ResultA slow change of the central mass(3.17)

M r = constant, \frac{δ r}{r} = - \frac{δ M}{M}

$M$	the mass of the central body, changing slowly
$r$	the radius of a circular orbit round it
$δ M, δ r$	small changes of the two

A sudden loss of mass is different. The orbiting body keeps its speed, which is now too fast for the weaker pull, and it moves onto an ellipse whose closest point is where it was. If the central body suddenly loses half of its mass or more, the speed reaches the new escape speed and the orbiting body leaves for good.

Common errorsSection 3.7

Kepler's third law gives the total mass of the two bodies. For a moon round a planet the moon's mass hardly matters, but for a binary star it is often a large fraction of the total, up to half.
The separation $a$ in Kepler's third law is the distance between the two bodies, $a_{1} + a_{2}$ . Using the distance of one body from the centre of mass gives a mass that is too small.
Momentum is conserved during a collision, and kinetic energy usually is not, since some of it heats and breaks the bodies. Energy conservation holds again for the orbit after the collision.
A slow loss of mass and a sudden one have different effects. A slow loss widens a circular orbit and keeps it circular, while a sudden loss leaves the body at the closest point of a new ellipse.

ExercisesSection 3.75 questions

3.7.1
The Moon has 1/81.30 of Earth's mass and orbits 384 400 km from Earth's centre. How far from Earth's centre is the centre of mass of the two?
km
3.7.2
Two stars go round each other once every 2 years, at a separation of 2 au. What is their total mass?
solar masses
3.7.3
Is the centre of mass of the Solar System always inside the Sun?
3.7.4
Charon has 1/8 of Pluto's mass, and their centres are 19 640 km apart. How far from Pluto's centre is their centre of mass? Pluto's radius is 1188 km.
km
3.7.5
A planet is on a circular orbit round a star. The star suddenly throws off exactly half of its mass. What happens to the planet?

Olympiad problemsPast IOAA problems, restated3 problems

2009How the Sun's loss of mass widens Earth's orbit10 points

Adapted from IOAA 2009 (Tehran), theory problem 12. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The Sun loses mass because of the nuclear reactions at its centre. How much does the radius of Earth's orbit grow in 100 years because of this? Assume that Earth's orbit stays circular. Take the Sun's luminosity as $3.828 \times 10^{26}$ W and its mass as $1.988 \times 10^{30}$ kg.

I solved it without the solution

AnswerEarth's orbit grows by about 1.0 m in 100 years, about 1 cm a year. The solar wind carries off about another third as much.

2015A planet's density from its moon10 points

Adapted from IOAA 2015 (Magelang), theory problem 2. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A moon of a planet has a period of 7 days, 3 hours and 43 minutes, and its semi-major axis is 15.3 times the mean radius of the planet. The Moon has a period of 27 days, 7 hours and 43 minutes and a semi-major axis 60.3 times Earth's mean radius. Ignore the masses of the moons next to their planets. What is the ratio of the planet's mean density to Earth's?

I solved it without the solution

AnswerThe planet's density is 0.238 times Earth's. The data fit Jupiter and its moon Ganymede, and a density a quarter of Earth's shows that Jupiter is mostly hydrogen and helium.

2023Pushing an asteroid's moon40 points

Adapted from IOAA 2023 (Katowice), theory problem 12. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The DART spacecraft struck Dimorphos, a small moon of the asteroid Didymos, to test how an impact changes its orbit. Before the impact Dimorphos moved on a circular orbit round Didymos with a period of 11.92 h. The masses of Dimorphos and Didymos are $m = 4.3 \times 10^{9}$ kg and $M = 5.6 \times 10^{11}$ kg. At impact the spacecraft had mass $m_{s} = 580$ kg and moved at $v_{s} = 6.1$ km/s relative to Dimorphos. Ignore all other bodies.

(a) Find the expected change of the period, in minutes, if the collision was head-on, central and perfectly inelastic, so that the spacecraft stayed in Dimorphos. [20 points]

(b) The period was in fact observed to change by $Δ P_{0} = - 33$ minutes, because the impact threw debris off Dimorphos, which carried momentum away. Find the momentum of the debris as a fraction of the momentum of Dimorphos before the impact. Take the mass of the debris to be much smaller than Dimorphos's. [15 points]

I solved it without the solution

Answer(a) The period should fall by about 10.0 minutes. (b) The debris carried away a momentum of 0.0107, about 1.1 per cent, of Dimorphos's momentum, more than twice the spacecraft's own. (c) Dimorphos slowed by 2.71 mm/s, 3.3 times as much as the spacecraft alone would have slowed it.