The observational laws

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

Kepler’s first law says that an orbiting planet moves in an ellipse with the Sun at one focus. A focus is one of two special points inside an ellipse; the Sun is usually not 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 T² ∝ a³, where T is orbital period (s) and a is the semi-major axis of the elliptical orbit (m). For the circular orbits used in IB calculations this becomes T² ∝ r³, where r is the orbital radius measured from the centre of the central body (m).

Circular models and real orbits

In this topic, calculations assume circular orbital motion unless the question is explicitly qualitative. Treat that as a model, not a claim that nature makes perfect circles. Real planets and comets often follow elliptical orbits; in an elliptical orbit the total energy stays constant, but kinetic energy and gravitational potential energy are continually exchanged as the separation changes.

Uniform circular motion resembles a real orbit because a central force keeps changing the direction of the velocity. The difference is that the circular model keeps radius and speed constant, while an elliptical orbit changes both.

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 work out 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 its picture of orbiting from astronomy: electrons were imagined as moving around the nucleus rather like planets around the Sun. In both cases the attraction is an inverse-square force, so the analogy is tempting. It breaks down because a classical orbiting electron is accelerating and should radiate energy continuously, spiralling into the nucleus. That failure gives 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 with a force. Its magnitude is proportional to both masses, and inversely proportional to the square of the distance between them:

F = G**m₁m₂ / r², where F is the gravitational force magnitude (N), G is the universal gravitational constant (N m² kg⁻²), m₁ and m₂ are the two masses (kg), and r is the separation between their centres of mass (m).

The force is always attractive. Each mass pulls on the other with the same magnitude of force, but in the opposite direction, matching Newton’s third law. The inverse-square part is the key detail: 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 times can be compared. G is small, which is why 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 decorations in the data booklet; they are part of the shared language that lets physicists calculate consistently.

From point masses to spherical masses

The formula above is for point masses. For a uniform sphere, or for any spherically symmetric mass distribution, the gravitational field outside the sphere is the same as if all its mass were concentrated at its centre. That 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 counts as an extended body, not as 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 consider points outside it. In that case, its mass behaves as though it were concentrated at its centre of mass.

A centre of mass is a point that represents the average position of the mass distribution of an object. A centre of gravity is a point through which the resultant weight of an object acts in a particular gravitational field. In a uniform gravitational field, these are the same point. 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 poor when the gravitational field varies appreciably across the object. Tides give the usual 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. The field model is useful: it lets us describe how a source mass affects space before any second mass is placed there.

Gravitational field strength is a vector quantity equal to the gravitational force per unit mass on a small test mass placed at a point. That test mass has to be small enough that it does not noticeably change the field being measured.

g = F / m, where g is gravitational field strength (N kg⁻¹) and m is the test mass (kg).

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

g = G**M / r², where M is the source mass producing the field (kg).

The direction of g is the direction of the force on the test mass. Around an isolated spherical mass, it points radially inward.

Resultant field strength

Gravitational field strength is a vector, so field strengths add vectorially. In this topic, resultant field strength calculations are restricted to points on the line joining two bodies. On that line, the two field contributions point either in the same direction or in opposite directions, so the calculation becomes 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 = m**a, where a is acceleration (m s⁻²). Combining this with g = F / m gives a = g for a freely falling object when gravity is the only force. 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 weakens 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: closer lines mean a stronger field. Around a point mass or a uniform spherical mass, the pattern is radial, and the line density increases 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. Line density should match field strength. For a single spherical planet, draw inward radial lines, closer together near the planet.

The same field language is useful beyond this topic. The ideas of source, test object, field strength, vector direction, field lines and potential reappear for 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 due to their positions in a gravitational field. For gravity on a large scale, we usually define it relative to infinite separation.

The gravitational potential energy of a system equals 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 definition works well because gravity is a long-range inverse-square interaction. At any finite separation, the masses still attract each other, but as separation tends to infinity the force tends to zero. We set 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 shows that energy must be supplied to separate the masses back to infinity.

Two-body potential energy

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

Eₚ = −G**m₁m₂ / r, where Eₚ is the gravitational potential energy of the two-body system (J).

Here, r is the separation between the centres of mass of the two bodies. The zero of Eₚ is set at infinite separation. As r increases, Eₚ becomes less negative and tends towards zero; as r decreases, Eₚ becomes more negative.

Orbit energy changes

Raising a satellite to a higher orbit increases its gravitational potential energy because the satellite–planet system is less tightly bound. To move it 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 is only an approximation in a nearly uniform field. For large changes in r, use the inverse-distance expression for Eₚ 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,

V₍g₎ = −G**M / r, where V₍g₎ is gravitational potential (J kg⁻¹).

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 Eₚ = m**V₍g₎.

It helps to think of gravitational potential as a well. A mass near a planet sits deep in that well. To move it upward toward zero potential at infinity, energy must be supplied. This picture of potential also lets us map gravitational fields without drawing a vector at every point.

Slope of the potential graph

Gravitational field strength tells you how quickly gravitational potential changes with position:

g = −ΔV₍g₎ / Δr, where ΔV₍g₎ is the change in gravitational potential between two nearby points (J kg⁻¹) and Δr is the small radial displacement between them (m).

The minus sign gives the direction. Gravitational potential increases as you move 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, you find the change in potential between two radii from the area under the field-strength curve, taking care with the signs.

Mapping fields using potential

Potential is a scalar, which often makes it easier to map than vector field strength. Once you have a potential map or graph, the field points “downhill” in potential, and its 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 with no change in kinetic energy, the work done by an external agent is

W = mΔV₍g₎, where W is the work done or energy transferred (J).

Use ΔV₍g₎ = V₍g,final₎ − V₍g,initial₎. If the final point is farther from the planet, the potential is less negative. So ΔV₍g₎ is positive, and work must be supplied.

For many launch-energy calculations, this equation is the neatest route. Say 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 to include the kinetic energy it has once it is moving in the circular orbit.

A real launch needs more fuel than the ideal mechanical-energy change suggests. Energy is also spent against atmospheric drag, in lifting fuel that is later burned, and because rockets do not transfer 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, provided its speed is unchanged, since ΔV₍g₎ = 0.

For a single isolated spherical mass, the equipotential surfaces are 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 useful local comparison: walking along a contour line involves no change in height, so gravitational potential energy does not change. Crossing contours corresponds to changing 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. That would contradict what equal potential means.

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

Energy condition for escape

Escape speed is the minimum speed an unpowered object must have at a point in a gravitational field to reach infinity with zero speed remaining.

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

v₍esc₎ = √(2G**M / r), where v₍esc₎ is escape speed (m s⁻¹).

You get this result by making the total mechanical energy zero: the positive kinetic energy exactly cancels the negative gravitational potential energy. The mass of the escaping object makes no difference. Energy conditions for escape at the same radius r from a mass M.

If the object is already moving in a parking orbit, it already has orbital kinetic energy, so it needs less extra energy than it would from rest at the same radius. Escape does not mean simply “get far away”; it means “make total mechanical energy at least zero”.

That is why energy gives a useful estimate of launch requirements. To escape a planet’s gravitational influence, a spacecraft must receive 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. The orbital speed is

v₍orbital₎ = √(G**M / r), where v₍orbital₎ is the speed needed for a circular orbit (m s⁻¹).

Measure r from the centre of the planet or star, not from its surface. So if a satellite is at height h above a planet of radius R, then r = R + h, where h is height above the surface (m) and R is the radius of the planet (m).

For circular orbits in this topic, combine circular motion with gravity to get the period relation: T² = 4π²r³ / (G**M). 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

Eₖ = GMm / (2r), where Eₖ is kinetic energy (J).

Combining this with Eₚ = −GMm / r gives the total mechanical energy

E = −GMm / (2r), where E is total mechanical energy (J).

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. The word “non-rotating” matters because otherwise the satellite already has kinetic energy due to the planet’s rotation.