Newtonian gravitational potential ($\Phi$)

The Newtonian gravitational potential $\Phi$ is potential energy per unit mass of a test particle. Its gradient determines the Newtonian gravitational field:In SI, $U$ has units $\mathrm{J}$, while $\Phi$ has units $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}={\mathrm{m}}^2{\mathrm{s}}^{-2}$; its gradient has units of acceleration, not force. Adding a constant to $\Phi$ leaves the Newtonian gravitational field unchanged. For an isolated point mass, the convenient reference is $\Phi (\infty )=0$. ^[ref]

Increasing the radius from $r_0$ to $r_1>r_0$ raises the Newtonian gravitational potential energy:An external agent supplies this positive work if the initial and final kinetic energies agree; gravity does work $-\Delta U$. For $r_1=2r_0$, the required work is $GMm/(2r_0)$. ^[ref]

For constant mass density $\rho$ on the lower half of the spherical shell $a\le |{\mathbf{r}}^{\prime }|\le b$, with $0<a<b$, the displayed Newtonian gravitational potential applies at $(0,0,h)$ for every $h>0$, with zero at infinity. Use source spherical coordinates with polar angle $\alpha \in [\pi /2,\pi ]$. For a fixed source radius $s$,Multiplication by $-2\pi G\rho s^2$ and integration from $a$ to $b$ leaves the antiderivative $s^3/3+h{s}^2/2-(s^2+h^2{)}^{3/2}/3$, proving the formula. The entire positive axis is outside the matter, even for $a<h<b$, so there are no interior/exterior radial cases. The limit at the origin is $-\pi G\rho (b^2-a^2)$; far away the leading term is $-GM/h$, with total mass $M=2\pi \rho (b^3-a^3)/3$.

The usual Newtonian gravitational potential satisfies $\mathbf{g}=-\nabla \Phi$ and ${\nabla }^2\Phi =4\pi G\rho$. An equally consistent convention sets $\phi =-\Phi$, giving $\mathbf{g}=\nabla \phi$ and ${\nabla }^2\phi =-4\pi G\rho$. With zero potential at infinity, a point mass has $\Phi =-GM/r$ and $\phi =GM/r$. Both conventions describe the same inward gravitational field and the same Gauss's law for gravity. Only the signs of the potential and its differences change. The gravitational potential energy per unit test mass is $\Phi$, or $-\phi$ in the alternative convention. A formula for a potential difference must therefore be read together with its relation to the gravitational field.

A sinusoidal surface-density component ${\Sigma }_k\cos (kx)$ on $y=0$, with $k>0$ and translational invariance in the remaining direction, produces the displayed gravitational potential. Laplace's equation away from the sheet separates into a cosine in $x$ and a decaying exponential in $|y|$. Continuity across the sheet excludes a potential jump, while integrating Poisson equation for Newtonian gravity across it gives $[{\Phi }_y{]}_{0^{-}}^{0^{+}}=4\pi G{\Sigma }_k\cos (kx)$. For a potential $C\cos (kx)e^{-k|y|}$ that derivative jump is $-2kC\cos (kx)$, giving $C=-2\pi G{\Sigma }_k/k$.

More generally a specified jump $J\cos (kx)$ gives potential $-J\cos (kx)e^{-k|y|}/(2k)$. Each spatial Fourier mode decays on the length scale $1/k$, so fine density structure has a shorter-range field. The sinusoid changes sign and is naturally interpreted as a density perturbation component. General sheet distributions are built by superposing their Fourier modes; decay alone does not restrict a harmonic potential to a single mode.

For radius $R$, constant mass density $\rho$ and $M=4\pi \rho R^3/3$, normalize the Newtonian gravitational potential to zero at infinity. ThenBy the spherical shell theorem, each spherical shell contributes the potential of a point mass outside it and a constant inside it. Summing inner and outer shells gives $\Phi (r)=-4\pi G\rho [r^2/3+(R^2-r^2)/2]$ inside. Equivalently, angular integration of the free-space Green function of the Laplacian gives ${\int }_{-1}^1(r^2+r^{\prime 2}-2r{r}^{\prime }\mu {)}^{-1/2}d\mu =[r+r^{\prime }-|r-r^{\prime }|]/(r{r}^{\prime })$. The resulting radial integral has the same value.

The potential and its first radial derivative are continuous at $R$. The interior gravitational acceleration is $-\nabla \Phi =-(GM/R^3)\mathbf{r}$, and ${\nabla }^2\Phi =4\pi G\rho$. At the centre $\Phi (0)=-3GM/(2R)$ while its gradient vanishes.

The dimensionless coefficient $J_2$ describes the leading axisymmetric departure of a planet's external Newtonian gravitational potential from a monopole: $\Phi =-G{M}_p/r+[G{M}_p{J}_2{R}_p^2/r^3]P_2(\cos \vartheta )$, where $\vartheta$ is colatitude from the spin axis. An oblate mass distribution has positive $J_2$. The disturbing function is minus the nonspherical potential term. On a circular inclined Kepler orbit, $\cos \vartheta =\sin I\sin M$, so orbit averaging gives ${\overline{\mathcal{R}}}_{J_2}=G{M}_p{J}_2{R}_p^2{a}^{-3}(1/2-3{\sin }^{2}I/4)$. The exact circular nodal rate is $-3n{J}_2(R_p/a{)}^2\cos I/2$; its small-tilt limit is $-3n{J}_2(R_p/a{)}^2/2$.

For $\Phi =-\mu /r+{\alpha }^2\mu (3z^2-r^2)/r^5$, a nearly circular equatorial orbit has $n^2=\mu /R^3$ and azimuthal frequency ${\Omega }_c^2=n^2(1+3{\alpha }^2/R^2)$. The radial epicyclic frequency is ${\kappa }^2=Rd{\Omega }_c^2/dR+4{\Omega }_c^2$, and the vertical epicyclic frequency is ${\nu }^2={\partial }_z^2\Phi {|}_{z=0}$. Radial stability requires $R^2>3{\alpha }^2$. The nodal precession rate ${\Omega }_c-\nu$ is negative and the apsidal precession rate ${\Omega }_c-\kappa$ positive. These sign statements concern small eccentricity and inclination. For an arbitrary inclined orbit, the averaged argument of periapsis instead evolves as $(3/2)n{\alpha }^2[5{\cos }^{2}i-1]/[a^2(1-e^2{)}^2]$ and can regress.

An axisymmetric logarithmic Newtonian gravitational potential has spheroidal equipotential surfaces and a flat galaxy rotation curve: $R{\Phi }_R(R,0)=V^2$. The singular model is defined away from the origin; a fixed reference length makes the logarithm dimensionless. Its central $r^{-2}$ mass density cusp is locally integrable, but its total mass diverges at large radius.

The Poisson equation for Newtonian gravity givesFor $V>0$, nonnegative mass density away from the origin is equivalent to $q^2\ge 1/2$. Strict positivity requires $q^2>1/2$. This condition does not by itself guarantee a nonnegative galactic distribution function.

A radial Newtonian gravitational potential $\Phi =-A{r}^{-\beta }$ produces inward acceleration $A\beta r^{-\beta -1}$. The point-mass case has $\beta =1$ and $A=GM$. Treating the power-law gravitational potential as prescribed does not require it to be sourced by the flowing gas.

For a binary star viewed far from its centre of mass, let $\mathbf{r}$ be the separation and $\mu$ the reduced mass, $M=M_1+M_2$ the total mass, $r=|\mathbf{r}|$ and $x=|\mathbf{x}|$. The Taylor expansion of the point-mass Newtonian gravitational potential has zero dipole because ${\sum }_A{M}_A{\mathbf{x}}_A=0$. Its second moment is ${\sum }_A{M}_A{x}_{Ai}{x}_{Aj}=\mu r_i{r}_j$, soThis holds for an instantaneous separation in any orbit, not only a circular orbit. The mass quadrupole moment convention $Q_{ij}=\mu (r_i{r}_j-r^2{\delta }_{ij}/3)$ gives ${\Phi }_2=-3G{Q}_{ij}{x}_i{x}_j/(2x^5)$.

A flattened power-law Newtonian gravitational potential has equipotential spheroids $R^2+{\lambda }^2{z}^2=\text{constant}$. For ${\Phi }_0<0$ and $0<\beta <1$, its midplane circular orbit angular frequency satisfies ${\Omega }^2=-\beta {\Phi }_0{R}_0^{\beta }{R}^{-\beta -2}$.

Applying the Poisson equation for Newtonian gravity to a flattened power-law gravitational potential yieldsFor $0<\beta <1$, nonnegative mass density everywhere outside the origin requires and is ensured by $\beta \le {\lambda }^2\le 2/(\beta +1)$. The central density singularity is locally integrable and carries no point mass, although the total mass diverges at large radius in this scale-free model.

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.

A spherically symmetric potential depends only on distance $r$ from one point. Its force is central, so angular momentum is conserved.