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

The Newtonian potential of a point mass $M$ at the origin isIt 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 $\Phi /(GM/r_0)=-1/z$; 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: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 $v_c=\sqrt{GM/r_0}$. Choosing units with $G=M=r_0=1$ gives $g_0=1$, $v_c=1$, and $v_{\mathrm{esc}}=\sqrt{2}\simeq 1.41421$. A radial launch at speed $1$ reaches $r=2$ and returns; the same speed directed tangentially gives a circular orbit. ^[ref]

A thin uniform spherical shell of fixed mass $M_s$ and radius $R$ has Newtonian gravitational potential energyThis follows from the spherical binding-energy mass integral: the enclosed mass is zero inside $R$ and $M_s$ outside. The radial generalized force, positive outward, is consequently $-{\partial }_R{W}_{\mathrm{self}}=-G{M}_s^2/(2R^2)$, taking the derivative at fixed $M_s$. It is an inward resistance to uniform expansion; individual surface forces cancel as vectors by spherical symmetry.

If an interior spherical component has mass $M_c$, the spherical shell theorem adds inward load $G{M}_c{M}_s/R^2$. The total inward load isFor a swept gas fraction $f$ of an original enclosed mass $M$, with $M_s=fM$ and $M_c=(1-f)M$, this becomes $G{M}^{2}f(1-f/2)/R^2$. Omitting shell self-gravity would give $f(1-f)$ instead. In an accreting shell, evaluate the instantaneous load at its current mass; differentiating an assigned $M_s(R)$ in the fixed-mass energy formula would also include the energy of newly acquired material.

The sphere $\rho =k^{\prime }/(2\pi Gr)$ for $r<r_0$ has $M(r)=k^{\prime }{r}^2/G$ and $\Phi (r)=-k^{\prime }(2r_0-r)$, with zero potential at infinity. Its finite central potential contrasts with the logarithmic central divergence of a truncated inverse-square density.

Consider a uniform-density cylinder with an axisymmetric radial Fourier mode $\eta =ϵe^{ikz}$. Its displaced boundary gives a first-order surface mass density $\rho \eta$, although the interior density perturbation vanishes. The gravitational Poisson equation reduces to Laplace equation away from the interface. Regularity internally and decay externally give ${\psi }_{\mathrm{in}}=A{I}_0(kr)e^{ikz}$ and ${\psi }_{\mathrm{out}}=B{K}_0(kr)e^{ikz}$. Continuity at $R$ and the derivative jump ${\psi }_{\mathrm{out}}^{\prime }-{\psi }_{\mathrm{in}}^{\prime }=4\pi G\rho ϵe^{ikz}$, together with $I_0(\kappa )K_0^{\prime }(\kappa )-I_0^{\prime }(\kappa )K_0(\kappa )=-1/\kappa$, 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.