The observational laws

Kepler’s laws are empirical rules for the motion of planets and other orbiting bodies. They came from careful astronomical data, rather than from a theory of force. Newton later showed that gravity leads to these same rules.

Kepler’s first law says that an orbiting planet follows an ellipse with the Sun at one focus. A focus is one of two special points inside an ellipse; the Sun usually isn’t at the geometrical centre. A circle is the special case where the two foci coincide.

Kepler’s second law says that the line from the Sun to the planet sweeps out equal areas in equal times. In practical terms, the planet moves faster when it is closer to the Sun and slower when it is farther away.

Kepler’s third law says that $T^2 \propto a^3$, where $T$ is orbital period $(\text{s})$ and $a$ is the semi-major axis of the elliptical orbit $(\text{m})$. In the circular orbits used for IB calculations, this becomes $T^2 \propto r^3$, where $r$ is the orbital radius measured from the centre of the central body $(\text{m})$.

Circular models and real orbits

In this topic, calculations assume circular orbital motion unless the question is explicitly qualitative. That’s a model, not a claim that nature draws perfect circles. Real planets and comets often follow elliptical orbits; in an elliptical orbit the total energy stays constant, while kinetic energy and gravitational potential energy continually trade places as the separation changes.

Uniform circular motion resembles a real orbit because a central force continually changes the direction of the velocity. It differs from many real orbits because the circular model keeps radius and speed constant, while both change in an elliptical orbit.

Binary star systems show another limit of the “small object around a fixed large object” picture. If the two stars have comparable masses, both orbit their common centre of mass. To determine the nature of the stars, astronomers use measurements such as orbital period, separation or angular separation with distance, radial speeds from spectral shifts, spectra for temperature and composition, and luminosity changes if the stars eclipse each other.

Models that work, and models that fail

The old planetary model of the atom borrowed the idea of orbiting from astronomy: electrons were pictured moving around the nucleus rather like planets around the Sun. The attraction is an inverse-square force in both cases, so the analogy is tempting. But it breaks down because a classical orbiting electron is accelerating and should radiate energy continuously, spiralling into the nucleus. That failure teaches a useful lesson: a model can be productive and still not be the final story.

The inverse-square force

Newton’s universal law of gravitation says that any two point masses attract one another. The size of the force is proportional to both masses and inversely proportional to the square of the distance between them:

The force is always attractive. Each mass pulls on the other with the same size of force, but in the opposite direction, as required by Newton’s third law. The inverse-square term is the key feature here: double the separation, and the force drops to one quarter of its original value.

Constants and measurement

A physical constant is a quantity with a fixed value within a theory, so measurements made in different places and at different times can be compared. $G$ is small, so gravitational forces between laboratory-sized objects are tiny. Physicists determine constants through carefully designed experiments and, in modern SI, some constants are also used to define units. Their values are not just entries in the data booklet; they are part of the shared language physicists use to calculate consistently.

From point masses to spherical masses

The formula above applies to point masses. For a uniform sphere, or any spherically symmetric mass distribution, the gravitational field outside the sphere is the same as if all its mass were concentrated at its centre. This is why we can use Earth’s mass at Earth’s centre when calculating the gravitational force on a satellite outside Earth.

When size can be ignored

An extended body is an object whose physical size is not negligible compared with the distances being considered. A planet, moon, asteroid or satellite is an extended body, not a mathematical point.

We can treat an extended body as a point mass if its size is small compared with its separation from other bodies. The same model works for a uniform or spherically symmetric sphere when we are considering points outside it. In those cases, its mass acts as though it were concentrated at its centre of mass.

A centre of mass is the point representing the average position of an object’s mass distribution. A centre of gravity is the point through which the resultant weight of an object acts in a particular gravitational field. In a uniform gravitational field, these two points are the same. If the field changes significantly across the object, they need not be exactly the same.

When the point-mass model breaks

The point-mass model becomes weak when the gravitational field varies appreciably across the object. Tides are the standard example. The Moon’s field is slightly stronger on the near side of Earth than on the far side, so different parts of Earth’s oceans experience different gravitational effects. That variation across an extended body is exactly what the point-mass approximation hides.

Field idea

A gravitational field is a region of space where a mass experiences a gravitational force. Describing the field is useful because we can say what a source mass does to the space around it before any second mass is placed there.

Gravitational field strength is a vector quantity: the gravitational force per unit mass on a small test mass at a point. The test mass has to be small enough that it doesn’t noticeably alter the field being measured.

Using Newton’s law for a point mass or a uniform spherical mass,

$g = GM / r^2$, where $M$ is the source mass producing the field (kg).

The direction of $g$ matches the direction of the force on the test mass. For an isolated spherical mass, it points radially inward.

Resultant field strength

Gravitational field strength is a vector, so fields add vectorially. In this topic, resultant field strength calculations are limited to points on the line joining two bodies. Along that line, the two field contributions point either the same way or in opposite directions, so the calculation reduces to signed addition of magnitudes.

Field strength and acceleration

Close to a planet’s surface, $g$ is also the acceleration of free fall. Newton’s second law gives $F = ma$, where $a$ is acceleration ($m\,s^{-2}$). Combine this with $g = F / m$, and for a freely falling object with gravity as the only force, $a = g$. That is why $N\,kg^{-1}$ and $m\,s^{-2}$ are equivalent units.

Near the surface of a massive body, the radial field can often be treated as uniform: over a small height change, the field lines are essentially parallel and equally spaced. On a planetary scale, though, the field is radial and becomes weaker with distance.

Reading field lines

A gravitational field line is a curve whose tangent shows the direction of the gravitational field at each point. In gravity, the arrows point toward mass, since gravity is attractive.

Line spacing shows field strength: lines packed closer together mean a stronger field. Around a point mass or a uniform spherical mass, the field lines are radial, with the line density increasing as you move inward.

Sketching field lines

When you sketch gravitational field lines, use two rules. Arrows point in the direction a small test mass would accelerate. The line density should match the field strength. For a single spherical planet, draw radial lines pointing inward, with the lines closer together near the planet.

This shared field language is useful beyond this topic. The same ideas — source, test object, field strength, vector direction, field lines and potential — appear again in electric and magnetic fields. Using consistent terminology reduces ambiguity when physicists move between different kinds of interaction.

Energy belongs to the system

Gravitational potential energy is energy stored in a system of masses because of where those masses are in a gravitational field. For gravity on a large scale, we usually define this energy relative to infinite separation.

The gravitational potential energy of a system is the work done to assemble the system from infinite separation of its components. “Infinity” is not somewhere we travel to; it is a reference condition where the gravitational interaction tends to zero.

This works well because gravity is a long-range inverse-square interaction. At any finite separation, the masses still attract each other. As the separation tends to infinity, the force tends to zero. We choose the gravitational potential energy at infinite separation to be zero.

For a bound gravitational system, bringing the masses together from infinity releases energy, so the final gravitational potential energy is negative. The negative sign is not a trick: it tells us that energy must be supplied to separate the masses back to infinity.

Two-body potential energy

For two point masses, or two uniform spherical masses treated as point masses, the gravitational potential energy is

The separation $r$ is measured between the centres of mass of the two bodies. The zero of $E_p$ is taken to be at infinite separation. Increase $r$, and $E_p$ becomes less negative, moving towards zero. Decrease $r$, and $E_p$ becomes more negative.

Orbit energy changes

Raising a satellite to a higher orbit increases its gravitational potential energy, since the satellite–planet system is less tightly bound. If it’s moved slowly from one radius to a larger radius, energy must be supplied. If it moves inward, energy is released from the gravitational field.

The familiar near-surface expression for a small height change works only as an approximation in a nearly uniform field. For large changes in $r$, use the inverse-distance expression for $E_p$ instead of treating $g$ as constant.

Potential is energy per unit mass

Gravitational potential at a point is a scalar quantity equal to the work done per unit mass in bringing a small mass from infinity to that point without changing its speed.

For a point mass or a uniform spherical mass,

We define gravitational potential as zero at infinity. Closer to an isolated mass, $V_g$ is negative, since gravity is attractive and the system is bound.

Potential and potential energy are not the same thing. Potential describes the field at a point; potential energy depends on the mass placed there as well. A mass in a gravitational potential has $E_p = mV_g$.

It can help to picture gravitational potential as a well. A mass near a planet sits deep in that well. You have to supply energy to move it upward toward zero potential at infinity. The same potential picture lets us map gravitational fields without drawing a vector at every point.

Slope of the potential graph

Gravitational field strength depends on how quickly gravitational potential changes with position:

The minus sign gives the direction. Gravitational potential increases outward, toward zero, while the gravitational field points inward, toward decreasing $r$.

On a graph of $V_g$ against $r$, the magnitude of $g$ is the magnitude of the gradient. On a graph of $g$ against $r$, the change in potential between two radii comes from the area under the field-strength curve, provided the signs are handled carefully.

Mapping fields using potential

Potential is a scalar, so it’s usually easier to map than vector field strength. From a potential map or graph, the field direction is “downhill” in potential, and the field magnitude comes from how steeply the potential changes with distance.

Work from potential difference

When a mass moves between two points in a gravitational field and its kinetic energy does not change, the work done by an external agent is

Use $\Delta V_g = V_{g,\text{final}} - V_{g,\text{initial}}$. If the final point is farther from the planet, the potential is less negative. So $\Delta V_g$ is positive, and work must be supplied.

This equation is often the neatest route through launch-energy calculations. For example, when a spacecraft moves from a planet’s surface to a parking orbit, the gravitational part of the energy requirement comes from the potential difference. To place the spacecraft in orbit, you also need the kinetic energy it has in the circular orbit.

A real launch needs more fuel than the ideal mechanical-energy change suggests. Some energy is lost to atmospheric drag, some goes into lifting fuel that will later be burned, and rockets do not convert chemical energy perfectly into useful mechanical energy.

Surfaces of equal potential

An equipotential surface is a surface where every point has the same gravitational potential. If a mass moves along one of these surfaces, no work is done against gravity, as long as its speed stays unchanged, because $\Delta V_g = 0$.

For a single isolated spherical mass, the equipotential surfaces form concentric spheres centred on the mass. Near a planet’s surface, where the field is approximately uniform, they are approximately horizontal planes.

Map contours give a good local analogy. Walking along a contour line involves no change in height, so gravitational potential energy does not change. Crossing contours, however, corresponds to a change in gravitational potential.

Right-angle relationship

Gravitational field lines meet equipotential surfaces at right angles. If they didn’t, part of the gravitational force would act along the surface, so a mass moving along an equipotential would gain or lose energy — which contradicts the idea of equal potential.

Close equipotential surfaces show that potential changes quickly with distance, so the gravitational field is strong. Wider spacing means a weaker field. A potential map therefore shows direction and strength: field lines cross equipotentials at right angles, point toward lower potential, and use spacing to show the magnitude.

Energy condition for escape

Escape speed is the minimum speed an unpowered object needs at a point in a gravitational field if it is to reach infinity with zero speed left.

For escape from a point at radius r from a mass M,

You get this by setting the total mechanical energy to zero: the positive kinetic energy exactly cancels the negative gravitational potential energy. The mass of the escaping object does not affect the result. Energy conditions for escape at the same radius r from a mass M.

An object already moving in a parking orbit has orbital kinetic energy, so it needs less extra energy than an object starting from rest at the same radius. Escape is not about simply “getting far away”; the condition is that the total mechanical energy is at least zero.

That is why energy gives a neat way to estimate launch requirements. To escape a planet’s gravitational influence, a spacecraft must be given enough energy to climb out of the planet’s potential well and, if needed, to increase its kinetic energy from orbital speed to escape speed.

Speed for a circular orbit

For a body moving in a circular orbit around a much larger mass, gravity supplies the centripetal force. Its orbital speed is

Be careful with $r$: it’s measured from the centre of the planet or star, not from the surface. If a satellite is at height $h$ above a planet of radius $R$, then

For circular orbits in this topic, combine circular motion with gravity to get the orbital period relation: $T^2 = 4\pi^2 r^3 / (GM)$. This is the circular-orbit version of Kepler’s third law for a small orbiting body around a much larger central mass.

Energy in a circular orbit

The kinetic energy in a circular orbit is

Combining this with $E_p = -GMm / r$ gives the total mechanical energy

Use this formula for energy changes when a satellite moves from one circular orbit to another. A higher circular orbit has a less negative total energy, so work must be done on the satellite to move it there.

For a satellite starting from rest on the surface of a non-rotating planet, the ideal energy needed to place it in a circular orbit is the final total orbital energy minus the initial gravitational potential energy at the surface. “Non-rotating” matters because otherwise the satellite already has kinetic energy due to the planet’s rotation.

Small drag, big long-term effect

Atmospheric drag is a resistive force produced when a satellite collides with gas particles in the upper atmosphere. The atmosphere may be extremely thin, but a low-orbit satellite can still lose mechanical energy bit by bit.

The slightly counter-intuitive part is this: when a satellite loses energy to a small viscous drag force, it drops into a lower orbit and its orbital speed increases. Its total mechanical energy becomes more negative, its orbital radius decreases, and since $v_{\text{orbital}} = \sqrt{GM / r}$, the speed for the new lower circular orbit is larger.

It’s worth being precise about the energy bookkeeping. As the orbit decays, gravitational potential energy decreases — it becomes more negative. Kinetic energy increases, though not by as much. The total mechanical energy still decreases, because energy has been transferred to the atmosphere as internal energy.

That’s why low-Earth satellites need occasional boosts. Without them, the radius falls, the satellite reaches denser air, drag increases, and the decay can become fast enough for re-entry and burn-up. Engineers can also make use of atmospheric drag on purpose, for example by increasing a spacecraft’s effective area to lower an orbit without using as much fuel.