Celestial Mechanics: Gravitational Potential and Potential Energy

CM01 described gravity as a force and as a gravitational field. The next step is to reformulate the same physics in terms of work and potential. This viewpoint is essential because celestial mechanics repeatedly uses energy to classify orbits, derive escape conditions, analyze turning points, and understand bound versus unbound motion.

For a point mass M, CM01 gave The Gravitational Field

(1)

CM02 will derive the scalar gravitational potential

(2)

and the potential energy of a mass m,

(3)

The connection among force, field, potential, and work will then be summarized by

(4)

(5)

and

(6)

These are not new laws added to Newtonian gravity. They are alternative mathematical descriptions of the same conservative interaction [1, 2, 3, 4].

1 Work done by a force

For a constant force, introductory mechanics often writes work as

(7)

Gravity in celestial mechanics is not constant. Its magnitude changes with separation, so the correct expression is the line integral

(8)

where C is the path followed by the particle.

For gravity produced by a point mass M at the origin,

(9)

Suppose the particle moves radially from r₁ to r₂. Then

(10)

Therefore

Wg	= ∫ r1r2 Fg ⋅ dr	(11)
	= ∫ r1r2 ⋅	(12)
	= −GMm∫ r1r2 .	(13)

Since

(14)

we obtain

Wg	= −GMmr1r2	(15)
	= GMm.	(16)

Thus

(17)

This sign contains the physics.

If the mass moves outward, then

(18)

so

(19)

and therefore

(20)

Gravity opposes the outward motion and does negative work.

If the mass falls inward, then r₂ < r₁ and

(21)

Gravity then does positive work on the particle.

2 Conservative forces and potential energy

A force is called conservative when the work done between two points depends only on the endpoints, not on the particular path between them.

For a conservative force, one defines potential energy U by

(22)

For gravity,

(23)

Using the point-mass work result,

U2 − U1	= −GMm	(24)
	= GMm.	(25)

Therefore

(26)

Potential energy increases when a mass is moved outward against gravity and decreases when the mass falls inward.

3 Only differences in potential energy are physically required

The equations of motion depend on forces, and forces depend on spatial derivatives of potential energy. Adding any constant C to U leaves the force unchanged:

(27)

Then

(28)

Thus the zero of potential energy is a convention.

For isolated gravitational systems, the most useful convention is

(29)

This choice is especially convenient in celestial mechanics because it cleanly distinguishes finite-separation bound states from the limiting state of complete separation.

4 Deriving the point-mass potential energy

Take the initial point to be at infinity and the final point to be at radius r:

(30)

With U(∞) = 0,

U(r) − U(∞)	= GMm	(31)
	= −.	(32)

Therefore

(33)

This negative sign is a direct consequence of the chosen zero at infinity.

At any finite separation,

(34)

As the masses are separated farther and farther,

(35)

As they move closer together,

(36)

becomes more negative.

5 Why negative gravitational potential energy is physically sensible

Imagine two masses initially separated infinitely far apart and at rest. Under the convention

(37)

they begin with zero gravitational potential energy.

Now allow them to fall toward one another. Gravity does positive work, increasing kinetic energy. Conservation of energy requires a corresponding decrease in potential energy. Therefore the finite-separation state must have

(38)

Conversely, separating the masses back to infinity requires positive external work. That work raises U from a negative value to zero.

The negative value therefore encodes the energy that must be supplied to completely separate an attractive gravitational system.

6 The gravitational potential Φ

Potential energy depends on both the source mass M and the test mass m:

(39)

Just as CM01 separated gravitational field from force, it is useful to separate the source-generated potential from the test mass.

Define gravitational potential by

(40)

For a point mass,

(41)

Then the potential energy of any test mass is simply

(42)

The gravitational potential belongs to the gravitational field generated by the source distribution. Potential energy belongs to the interaction between that field and a particular mass placed within it.

This distinction parallels the relation

(43)

The two pairs are therefore

(44)

and

(45)

7 Units of gravitational potential

Since

(46)

its SI units are

(47)

Using

(48)

we obtain

(49)

These are the same dimensions as speed squared. This becomes important later because orbital energy per unit mass naturally contains terms such as

(50)

8 The shape of the point-mass potential

The function

(51)

approaches zero from below as r becomes large.

Its derivative is

(52)

Thus Φ increases as one moves outward. Moving inward takes the particle toward increasingly negative potential.

9 Differential relation between work and potential energy

For an infinitesimal displacement,

(53)

For a conservative force,

(54)

Therefore

(55)

A scalar field U(r) changes under an infinitesimal displacement according to

(56)

Equating the two expressions gives

(57)

Because this must hold for arbitrary infinitesimal displacements,

(58)

This is the fundamental relation between a conservative force and its potential energy.

10 From gravitational potential to gravitational field

Use

(59)

and

(60)

Then

mg	= −∇(mΦ)	(61)
	= −m∇Φ.	(62)

Cancel the test mass:

(63)

This is one of the most important relations in gravitational physics.

The field points in the direction in which gravitational potential decreases most rapidly.

11 Checking g = −∇Φ for a point mass

For spherical symmetry,

(64)

so

(65)

With

(66)

we have

(67)

Therefore

g	= −∇Φ	(68)
	= −r.	(69)

Hence

(70)

which is exactly the gravitational field obtained directly from Newton’s law in CM01.

The potential description therefore reproduces the original force law.

12 Equipotential surfaces

An equipotential surface is a surface on which

(71)

For a point mass,

(72)

so constant potential means constant radius. The equipotential surfaces are therefore spheres centered on the mass.

Along an equipotential surface,

(73)

But

(74)

Therefore a displacement tangent to an equipotential surface is perpendicular to ∇Φ and hence also perpendicular to g.

Thus

(75)

13 Work written directly in terms of potential

Because

(76)

we have

(77)

Since

(78)

then

(79)

Equivalently,

(80)

For a point mass,

Wg	= −m	(81)
	= GMm,	(82)

which agrees with the direct force integral.

14 Path independence

Once

(83)

has been established, the path independence of gravitational work follows immediately.

For any path from point 1 to point 2,

Wg	= m∫ 12g ⋅ dr	(84)
	= −m∫ 12∇Φ ⋅ dr	(85)
	= −m(Φ2 − Φ1).	(86)

Only the endpoints appear in the final result.

Therefore the work around any closed path is zero:

(87)

This is the operational statement that Newtonian gravity is conservative away from singular point sources.

15 Potential from several point masses

Gravitational fields obey superposition. Because gradients are linear, gravitational potentials do as well.

Suppose masses M_i are located at positions r_i. The potential at field point r is

(88)

Then

(89)

Potential is often easier to superpose than force because it is a scalar. One adds numbers rather than vectors, and takes the gradient only after the total scalar potential has been assembled.

This becomes extremely useful in multi-body celestial mechanics and in gravitational modeling of extended bodies.

16 Potential of a continuous mass distribution

Let a mass density be

(90)

An infinitesimal source element has mass

(91)

Its contribution to the potential at r is

(92)

Integrating over the source gives

(93)

The field is then obtained from

(94)

This scalar-potential formulation is one of the gateways from elementary celestial mechanics to geophysics, stellar structure, galactic dynamics, and numerical gravity models.

17 Mechanical energy conservation

The work-energy theorem states

(95)

If gravity is the only force doing work,

(96)

But

(97)

Therefore

(98)

so

(99)

Hence the total mechanical energy

(100)

is constant for motion under time-independent Newtonian gravity alone.

For a test mass m moving in the field of a fixed point mass M,

(101)

Dividing by m gives the specific mechanical energy

(102)

This equation will become central later when the course classifies circular, elliptic, parabolic, and hyperbolic trajectories.

18 Example 1: Earth’s gravitational potential at the surface

Using

(103)

and

(104)

Earth’s surface gravitational potential is approximately

Φ(RE)	= −	(105)
	≈−6.26 × 107 J/kg.	(106)

Therefore

(107)

This number says that, under the point-mass/spherical-Earth approximation, approximately 62.6 MJ of energy per kilogram would be required to raise matter from Earth’s surface to rest at infinite distance.

It does not mean that a launch vehicle must literally supply exactly that amount as usable propulsion energy; real launch problems include kinetic-energy requirements, atmosphere, finite burn trajectories, losses, Earth rotation, and other effects. Here the number is purely the gravitational potential-energy difference.

19 The near-surface approximation and mgh

Introductory mechanics often uses

(108)

This is not a different gravitational theory. It is the small-height approximation to the exact Newtonian potential.

Suppose an object of mass m is raised from radius R to radius R + h. The exact potential-energy change is

ΔU	= U(R + h) − U(R)	(109)
	= − +	(110)
	= GMm.	(111)

Thus

(112)

Combine the fractions:

(113)

If

(114)

then

(115)

so

(116)

At the surface,

(117)

Therefore

(118)

The familiar constant-g expression is therefore the first-order local approximation to the exact inverse-radius potential.

20 Example 2: raising one kilogram to 400 km altitude

Take

(119)

(120)

and

(121)

For m = 1 kg, the exact increase is

ΔU	= GME	(122)
	≈ 3.70 × 106 J.	(123)

Thus

(124)

Using constant surface gravity instead gives

ΔUmgh	≈ (9.82)(4.00 × 105)	(125)
	≈ 3.93 × 106 J/kg.	(126)

The constant-g estimate is about 6% high because Earth’s gravitational field weakens appreciably over a 400 km rise.

This example shows exactly when celestial-mechanics gravity must replace the local constant-g approximation.

21 Preview: escape speed from gravitational potential

Potential energy gives a short derivation of escape speed.

Suppose a particle starts at radius r with speed v_esc and just barely reaches infinity with zero final speed. Then

(127)

Initially,

(128)

At infinity,

(129)

so

(130)

Therefore

(131)

Cancel m and solve:

(132)

At Earth’s surface this gives approximately

(133)

The cancellation of the test mass is another expression of the equivalence between gravitational field and acceleration in Newtonian mechanics.

A more complete interpretation of this result belongs later in the orbital-energy lessons, where zero, negative, and positive specific energies will be connected to parabolic, elliptic, and hyperbolic motion.

22 Spherically symmetric bodies and the point-mass exterior potential

A major result developed more fully in later gravity examples is Newton’s shell theorem. For a spherically symmetric body of total mass M, the gravitational field outside the body is the same as if all mass were concentrated at the center:

(134)

With the convention Φ(∞) = 0, the exterior potential is therefore

(135)

This is why point-mass formulas work so well for planets and stars when we are outside a nearly spherical source.

Inside an extended body, however, the potential generally depends on the internal mass distribution. The simple exterior expression cannot automatically be continued through the interior.

23 Potential versus field: which description should we use?

The field formulation is

(136)

The potential is a scalar, while the field is a vector.

Potential is often advantageous when:

• adding contributions from many sources;
• using energy conservation;
• finding equilibrium points;
• exploiting symmetry;
• deriving orbital turning points;
• constructing effective potentials;
• solving gravitational boundary-value problems.

The field is often advantageous when:

• computing instantaneous acceleration;
• integrating equations of motion directly;
• determining local force direction;
• propagating trajectories numerically.

Celestial mechanics routinely moves between the two descriptions.

24 A compact hierarchy of gravitational quantities

For a point source mass M, the sequence is

(137)

followed by

(138)

A test mass m then has

(139)

and

(140)

The work done by gravity between two locations is

(141)

This compact network of relations is worth learning as a connected structure rather than as four independent formulas.

25 Common mistakes

• Forgetting that the zero of gravitational potential is conventional. The standard isolated-system choice is Φ(∞) = 0.
• Treating negative potential energy as an error. With zero at infinity, attractive point-mass gravity gives negative U at finite separation.
• Confusing potential Φ with potential energy U. They are related by U = mΦ.
• Confusing field g with force F. They are related by F = mg.
• Missing the minus sign in g = −∇Φ. Gravity points toward decreasing potential.
• Using mgh over distances that are not small compared with the source radius.
• Assuming that Φ = −GM∕r is valid everywhere inside an extended body. It is exact outside a spherically symmetric body but the interior potential depends on the mass distribution.
• Forgetting that gravitational work is positive during inward fall and negative during outward motion.
• Using only the magnitude GM∕r² when a vector field direction is required.

26 What CM02 adds to the series

CM01 introduced gravity through force and field. CM02 adds the scalar potential description and the energy viewpoint.

The central derivation chain is

(142)

Dividing by the test mass gives

(143)

Differentiating recovers the field:

(144)

The energy equation

(145)

now provides the bridge to orbital mechanics. The next lessons can use this framework to examine gravitational potentials of spherical bodies and then move into Newton’s orbital equation of motion and central-force dynamics.

References