Effective Potential and Radial Motion in Central Gravity

Effective Potential and Radial Motion in Central Gravity

The gravitational two-body problem is a central force problem.

Once angular momentum is conserved, the planar motion can be separated into:

• radial motion, which changes the separation r,
• angular motion, which changes the polar angle 𝜃.

The effective-potential method absorbs the angular part of the kinetic energy into an effective radial potential.

This reduces the orbital problem to a one-dimensional energy equation for

(1)

The method is useful for:

• locating periapsis and apoapsis,
• identifying allowed and forbidden radial regions,
• understanding the angular-momentum barrier,
• finding circular orbits,
• testing circular-orbit stability,
• distinguishing radial from nonradial motion.

1 Velocity in plane polar coordinates

For planar motion,

(2)

Therefore

(3)

The kinetic energy is

(4)

2 Angular momentum in a central force

For a central force,

(5)

Therefore angular momentum is conserved.

For planar motion,

(6)

Define the specific angular momentum

(7)

Thus

(8)

3 Energy with angular momentum

The gravitational mechanical energy is

(9)

Use

(10)

Then

(11)

Therefore

(12)

The entire radial problem is now contained in one equation.

4 The effective potential

Define

(13)

Then

(14)

The effective potential contains two terms:

(15)

and

(16)

The first term is positive and diverges as

(17)

when

(18)

The second term is the ordinary gravitational potential energy.

5 Specific effective potential

Divide the energy equation by

(19)

Using

(20)

and

(21)

we obtain

(22)

where

(23)

The units of

(24)

are

(25)

the same as specific orbital energy.

6 The angular-momentum barrier

When

(26)

(27)

as

(28)

Thus

(29)

This is often called the centrifugal barrier or angular-momentum barrier.

It is not an additional physical gravitational interaction.

It arises because conservation of angular momentum forces the transverse speed

(30)

to grow as the radius decreases.

Consequently the transverse kinetic energy grows as

(31)

7 Radial kinetic energy

From

(32)

the radial kinetic energy per unit mass is

(33)

Therefore

(34)

Since

(35)

the allowed radial region must satisfy

(36)

Regions where

(37)

are forbidden by energy conservation.

8 Radial turning points

A radial turning point occurs when

(38)

Therefore

(39)

Substituting,

(40)

Multiply by

(41)

(42)

For

(43)

the roots are

(44)

Only positive real roots are physical radii.

9 Bound nonradial motion

For

(45)

and negative energy above the effective-potential minimum, there are two positive turning radii:

(46)

The smaller radius

(47)

is periapsis.

The larger radius

(48)

is apoapsis.

The radial coordinate oscillates between these values:

(49)

At each radial turning point,

(50)

but the transverse speed is generally nonzero.

10 Zero-energy turning point

For

(51)

the turning-point equation becomes

(52)

Thus the finite radial turning point is

(53)

The other radial limit is

(54)

This is the parabolic escape threshold.

11 Positive-energy motion

For

(55)

there is one positive finite turning point.

It is the periapsis of an unbound trajectory.

As

(56)

(57)

so

(58)

approaches a nonzero value after the transverse speed has fallen toward zero.

The complete orbit is hyperbolic in the ideal inverse-square problem.

12 Circular orbit as an effective-potential equilibrium

A circular orbit has constant radius:

(59)

Therefore it occurs at an extremum of the effective potential:

(60)

Differentiate:

(61)

Set this equal to zero:

(62)

Hence

(63)

In terms of total angular momentum,

(64)

13 Circular-orbit speed from the effective potential

Since

(65)

the circular condition

(66)

gives

(67)

Therefore

(68)

The effective-potential method therefore reproduces the circular-orbit condition derived directly from centripetal acceleration.

14 Energy of the circular orbit

Evaluate the effective potential at

(69)

Using

(70)

Φeff(rc)	= −	(71)
	= −.	(72)

Therefore

(73)

This is the minimum possible energy for a fixed nonzero

(74)

15 Stability of the circular orbit

An extremum is stable if the effective potential has a local minimum.

Differentiate again:

(75)

At

(76)

with

(77)

r c	= −	(78)
	= > 0.	(79)

Thus the Newtonian circular orbit is stable against sufficiently small radial perturbations at fixed angular momentum.

16 Small radial oscillations about a circular orbit

Let

(80)

Near the minimum,

(81)

The radial motion is approximately harmonic:

(82)

with

(83)

The circular orbital angular frequency also satisfies

(84)

Therefore

(85)

for Newtonian inverse-square gravity.

This equality is closely related to the closed nature of Keplerian ellipses.

17 Radial equation of motion

The effective potential also gives the radial force equation.

Since

(86)

the effective radial force is

(87)

Differentiate:

(88)

Therefore

(89)

Divide by

(90)

(91)

18 Relation to the polar-coordinate acceleration equation

The radial acceleration in polar coordinates is

(92)

Newton’s radial equation is

(93)

Using

(94)

we recover

(95)

Thus the effective-potential equation is not introducing a new force law.

It is a reformulation of the original Newtonian equations using the conserved angular momentum.

19 Changing angular momentum changes the barrier

For fixed central mass, the effective potential is

(96)

Increasing

(97)

raises the small-radius barrier.

It also moves the circular-orbit minimum outward:

(98)

Thus larger angular momentum makes close approach more difficult for a trajectory with the same energy.

20 Special case: zero angular momentum

If

(99)

then

(100)

There is no angular-momentum barrier.

The energy equation becomes

(101)

Pure radial infall can therefore reach

(102)

in the ideal point-mass model.

The singularity reflects the idealized Newtonian point source.

For a physical extended body, the interior gravitational field must replace the point-mass field.

21 Radial versus nonradial turning points

For

(103)

a turning point with

(104)

also has total speed

(105)

For

(106)

a radial turning point has

(107)

but

(108)

Therefore

(109)

for nonradial motion.

This is why periapsis and apoapsis are not points where the orbiting body stops.

22 Example 1: Earth orbit from a turning point

Take Earth’s gravitational parameter as

(110)

Suppose at

(111)

the velocity is purely tangential with magnitude

(112)

Because

(113)

this radius is a radial turning point.

The specific angular momentum is

h	= r0v0	(114)
	= 5.60 × 1010 m2∕s.	(115)

The specific energy is

𝜀	= v02 −	(116)
	≈−2.494 × 107 J∕kg.	(117)

Since the energy is negative, the orbit is bound.

23 Example 1 continued: the second turning point

The turning-point equation is

(118)

One root is the known turning radius

(119)

The second positive root is

(120)

Thus the radial coordinate oscillates between approximately

(121)

and

(122)

Because the tangential speed at

(123)

is larger than the circular speed there, that turning point is the periapsis.

24 Example 2: circular orbit for a specified angular momentum

Suppose

(124)

about Earth.

Then

rc	=	(125)
	≈ 6.27 × 106 m.	(126)

The circular specific energy is

𝜀c	= −	(127)
	≈−3.18 × 107 J∕kg.	(128)

For that fixed

(129)

no motion is possible with

(130)

25 Dimensionless form

It is useful to normalize radius by the circular-orbit radius

(131)

Define

(132)

Since

(133)

the effective potential becomes

Φeff

= −.

(134)

Thus

(135)

The minimum occurs at

(136)

with value

(137)

This universal curve describes all Newtonian gravitational effective potentials with nonzero angular momentum after suitable scaling.

26 Discriminant and existence of turning points

The quadratic turning-point equation has discriminant

(138)

Real turning points require

(139)

Therefore

(140)

But

(141)

is exactly the minimum circular-orbit energy.

Thus the algebraic condition for real turning points is the same as the effective-potential requirement that the energy lie at or above its minimum.

27 Three energy regimes at fixed nonzero angular momentum

For fixed

(142)

• At the minimum energy, the two turning points coincide and the orbit is circular.
• For negative energy above the minimum, there are two finite turning radii and the motion is bound.
• For zero energy, there is one finite periapsis and the outer limit is infinity.
• For positive energy, there is one finite periapsis and unbound outward motion is allowed.

The effective-potential graph displays all four cases without solving the full orbit equation.

28 What the effective potential does not determine

The effective potential determines the radial dynamics for a given

(143)

and

(144)

It does not by itself give

(145)

or the geometric shape of the orbit.

To obtain the trajectory in the orbital plane, one must combine:

• angular-momentum conservation,
• the radial dynamics,
• the relation between r and 𝜃.

The next articles develop that connection.

29 Common mistakes

1. Treating the angular-momentum term as a new physical gravitational force.
2. Forgetting that U_eff is used for radial motion at fixed angular momentum.
3. Confusing total angular momentum L with specific angular momentum h = L∕m.
4. Using h²∕(2r) instead of h²∕(2r²).
5. Forgetting the negative sign of the gravitational term.
6. Allowing motion in a region where 𝜀 < Φ_eff.
7. Assuming a radial turning point means total speed is zero.
8. Forgetting that periapsis and apoapsis have nonzero tangential speed when h≠0.
9. Using the quadratic turning-point formula without checking for positive real roots.
10. Forgetting that a circular orbit occurs at an extremum of the effective potential.
11. Finding the extremum but failing to test stability with the second derivative.
12. Assuming the effective potential has a small-radius barrier when h = 0.
13. Using the radial-collision result for a physical extended body all the way to r = 0.
14. Confusing the circular-orbit radius h²∕μ with the turning radius for arbitrary energy.
15. Assuming the effective potential alone gives the full orbit shape r(𝜃).

30 Practice exercises

1. Derive the polar-coordinate kinetic energy for planar motion.
2. Starting from angular-momentum conservation, derive the effective potential.
3. Derive the specific effective potential in terms of h and μ.
4. Show that radial motion is allowed only where 𝜀 ≥ Φ_eff.
5. Derive the quadratic equation for radial turning points.
6. Analyze the number of positive turning points for negative, zero, and positive energy.
7. Derive the circular-orbit radius r_c = h²∕μ.
8. Derive the circular-orbit energy 𝜀_c = −μ²∕(2h²).
9. Show that the circular orbit is stable by evaluating the second derivative.
10. Derive the small-radial-oscillation frequency about a circular orbit.
11. Recover the radial equation r = h²∕r³ − μ∕r² from U _eff.
12. Explain why no angular-momentum barrier exists for h = 0.
13. Reproduce the Earth turning-point example with r₀ = 7000 km and v₀ = 8.00 km/s.
14. Reproduce the circular-orbit example for h = 5.00 × 10¹⁰ m²∕s.
15. Derive the dimensionless effective-potential curve 1∕(2x²) − 1∕x.

31 Summary

For planar motion in Newtonian central gravity,

(146)

The conserved energy can be written

(147)

with

(148)

Per unit mass,

(149)

Allowed radial motion requires

(150)

Radial turning points satisfy

(151)

The circular orbit occurs at the effective-potential minimum:

(152)

with

(153)

For Newtonian inverse-square gravity,

(154)

so the circular orbit is radially stable.

The effective-potential method converts a two-dimensional central force problem into a one-dimensional radial-energy problem while preserving the influence of angular momentum.

References

References