Newtonian potential of a point mass ( $Φ (r) = - GM / r$ )

The Newtonian potential of a point mass

M

at the origin is

Φ (r) = - \frac{GM}{r}, r > 0, Φ (\infty) = 0.

It is the free-space Green function of the Laplacian with the normalization of the Poisson equation for Newtonian gravity. Differentiating gives inward gravitational acceleration

- GM / r^{2}

. With

z = r / r_{0}

, the normalized Newtonian gravitational potential is

Φ/ (GM / r_{0}) = - 1/ z

; at

z = 1, 2, 4

it equals

- 1, - 1/2, - 1/4

.

For a radially outward launch from

r_{0}

, conservation of mechanical energy in the fixed-source test particle approximation gives the minimum escape velocity:

\frac{E}{m} = \frac{v _{0}^{2}}{2} - \frac{GM}{r _{0}} = 0 ⟹ v_{esc} = \frac{2 GM}{r _{0}} .

At this threshold the speed tends to zero only at infinite separation. The model neglects air resistance, external tidal forces, and general relativity. ^[ref]

A circular orbit at

r_{0}

instead requires tangential speed

v_{c} = GM / r_{0}

. Choosing units with

G = M = r_{0} = 1

gives

g_{0} = 1

,

v_{c} = 1

, and

v_{esc} = 2 ≃ 1.41421

. A radial launch at speed

1

reaches

r = 2

and returns; the same speed directed tangentially gives a circular orbit. ^[ref]

Figure 1.
Energy balance and outward radial speeds below, at, and above escape
. Left: at the escape velocity, positive kinetic energy exactly balances negative Newtonian gravitational potential energy. Right: radial launches at normalized speeds $1$ , $2$ , and $1.6$ respectively turn at $r / r_{0} = 2$ , approach zero speed at infinity, and retain a positive limiting speed. The plotted trajectories use a fixed source and the test particle approximation. Details.

Table of contents

Self-gravity in a thin spherical shell expansion ( $W_{self} = - G M_{s}^{2} / (2 R)$ )

A shell of fixed mass

M_{s}

R

has self-energy

- G M_{s}^{2} / (2 R)

. Its inward generalized force opposing an increase of radius is

G M_{s}^{2} / (2 R^{2})

. Including this in a swept-gas shell threshold changes an interior-only factor

f (1 - f)

to

f (1 - f /2)

when the original total mass is held fixed.

Gravitational potential of a truncated inverse-radius sphere ( $Φ (r) = - k^{'} (2 r_{0} - r)$ )

ρ = k^{'} / (2 π G r)

for

r < r_{0}

has

M (r) = k^{'} r^{2} / G

and

Φ (r) = - k^{'} (2 r_{0} - r)

, with zero potential at infinity. Its finite central potential contrasts with the logarithmic central divergence of a truncated inverse-square density.

Axisymmetric gravitational potential of a displaced cylindrical interface ( $ψ_{in} = - 4 π GρR K_{0} (k R) I_{0} (k r) η$ )

Consider a uniform-density cylinder with an axisymmetric radial Fourier mode

η = ϵ e^{ik z}

. Its displaced boundary gives a first-order surface mass density

ρ η

, although the interior density perturbation vanishes. The gravitational Poisson equation reduces to Laplace equation away from the interface. Regularity internally and decay externally give

ψ_{in} = A I_{0} (k r) e^{ik z}

and

ψ_{out} = B K_{0} (k r) e^{ik z}

. Continuity at

R

and the derivative jump

ψ_{out}^{'} - ψ_{in}^{'} = 4 π Gρ ϵ e^{ik z}

, together with

I_{0} (κ) K_{0}^{'} (κ) - I_{0}^{'} (κ) K_{0} (κ) = - 1/ κ

, give the title amplitude. The prime differentiates the modified Bessel function argument. This axial mode is different from an azimuthal mode on a gravitational potential of a displaced cylindrical interface.

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