2.1. Matter and the gravitational field¶
A theory for the motion of objects under the influence of gravity requires two ingredients: how mass gives rise to a gravitational force and how objects move under the influence of this force. In classical physics, the laws for each of these go back to Newton and contemporaries: Newton’s law of universal gravitation that gives the force \(F\) from a mass \(M\) acting on a second mass \(m\) in terms of the distance \(r\) between the two as \(F = -GMm/r^2\), and Newton’s second law that gives the relation between force and acceleration \(a\) for an object with mass \(m\) as \(F = ma\). A more modern understanding of the law of universal gravitation, however, emphasizes that the three-dimensional force \(\vec{F}(\vec{x})\) derives from a scalar-valued gravitational potential \(\Phi(\vec{x})\) through: \begin{equation}\label{eq-force-gradient-potential} \vec{F}(\vec{x}) = -m\,\nabla \Phi(\vec{x})\,, \end{equation} and replaces Newton’s inverse-square law of gravitation with the Poisson equation as the fundamental equation relating mass and gravitational force: \begin{equation}\label{eq-sphergrav-poisson} \nabla^2 \Phi(\vec{x}) = 4\pi\,G\,\rho(\vec{x})\,, \end{equation} where \(\rho(\vec{x})\) is the mass density, \(G\) is the gravitational constant, and \(\nabla^2\) is the Laplace operator.
The reason that the Poisson equation is more properly considered to be the fundamental equation between mass and gravitational force is that it is the direct Newtonian limit of Einstein’s field equation \(R_{\mu\nu} -R\,g_{\mu\nu}/2 = 8\pi G\,T_{\mu\nu}\) in the general theory of relativity, which is our best theory of gravity so far. This is not a book about the general theory of relativity and the origin of Einstein’s field equation is typically not important for understanding galaxy dynamics, but a key point is that Einstein’s field equation reduces to the Poisson equation in the limit that velocities \(v\) and the gravitational potential are small compared to the speed of light \(c\) (the potential \(\Phi\) has units of velocity squared, so this limit is \(|\Phi|/c^2 \ll 1\) and \(v/c\ll 1\)). This limit always applies on the scale of galaxies, with the notable exception of gravitational lensing by galaxies as discussed in Chapter 15. For the interested reader, Appendix C has a self-contained discussion of Einstein’s field equation and how it reduces to the Poisson equation in the low-velocity limit.
Before continuing our discussion, it is worth noting that the Poisson equation (or its generalization, Einstein’s field equation) is a hypothesis. Like any other physical theory, we test the Poisson equation by making predictions derived using this equation and testing these predictions against observational data. Gravity as defined by the Poisson equation is extraordinarily well tested using laboratory experiments and solar-system gravitational dynamics: it is known to hold to one-part-in-one-billion in these settings (Adelberger et al. 2003; Will 2014). Partly because of these tests, most astrophysicists assume that the Poisson equation holds on the scales of galaxies and on the scale of the Universe (as Einstein’s equation) and that any anomalies that may result from its application are due to new forms of mass or energy that enter the mass density \(\rho\) or the energy-momentum tensor \(T_{\mu\nu}\), rather than indicating a problem with the Poisson equation itself. Large anomalies are indeed known to exist: these are dark matter, which is a \(\approx100\%\) anomaly on the scales of galaxies, and dark energy, which is a \(\approx100\%\) anomaly on the largest cosmological scales (meaning that ignoring them leads to \(\mathcal{O}(1)\) discrepancies with the observations).
Throughout this book, we will assume that the Poisson equation holds and, thus, that dark matter (and dark energy, but it is less important for our purposes) is a new form of matter whose density distribution can be studied using the Poisson equation. Theories that attempt to account for the anomalies typically interpreted as dark matter and dark energy without introducing new forms of matter and energy do so by modifying the Poisson equation or Einstein’s equation. Such modifications have to account for the fact that the Poisson equation holds to high accuracy in the solar system and, thus, can only change the Poisson equation in vastly different physical regimes.
Starting from the Poisson equation, we can derive Newton’s inverse-square law by computing the gravitational potential for a point mass \(M\) at position \(\vec{x}\). Because the Laplacian is a differential operator, we can move the origin of the coordinate system to any position, and calculations are most straightforward if we position the mass at the origin. The density in this case is \(\rho = M\,\delta(\vec{x})\), where \(\delta(\cdot)\) is Dirac’s delta function (Chapter B.3.2) . To show that the potential \(\Phi(\vec{x}) = -GM/r\), where \(r = |\vec{x}|\), is the solution of the Poisson equation in this case, we compute the Laplacian of \(-GM/r\) for \(r \neq 0\) using the expression for the Laplacian in spherical coordinates (Equation A.8): \begin{align} \nabla^2 \left(\frac{-GM}{r}\right) & = -GM\,\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\,\frac{\partial}{\partial r}\left[\frac{1}{r}\right]\right) = GM\,\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\,\frac{1}{r^2}\right) = 0\,. \end{align} Thus, the density is zero for all \(r\neq 0\). Finally, to show that at \(r=0\), the density corresponding to \(\Phi(\vec{x}) = -GM/r\) is the correct delta function, we integrate the Laplacian of \(\Phi/(4\pi G)\) over a small spherical volume \(V\) of radius \(R\) with surface \(S\), because this should equal the mass \(M\): \begin{align} \int_V \mathrm{d}V\, \nabla^2\left(-\frac{GM}{4\pi G\,r}\right) & = -\frac{M}{4\pi}\,\int_V \mathrm{d} V\, \nabla\cdot\left(\nabla\frac{1}{r}\right)\\ & = -\frac{M}{4\pi}\,\int_S \mathrm{d} S\, \vec{\hat{r}}\cdot \nabla\frac{1}{r}\\ & = -\frac{M}{4\pi}\,\int_S \mathrm{d} S\, \vec{\hat{r}}\cdot \vec{\hat{r}}\,\frac{\partial}{\partial r}\left(\frac{1}{r}\right)\\ & = 4\pi\,\frac{M}{4\pi}\,\frac{R^2}{R^2} = M\,. \end{align} Here, we have used the divergence theorem from Equation (B.5) in going to the second line and the expression for the gradient in spherical coordinates in going to the third line (Equation A.7); \(\vec{\hat{r}}\) is the unit vector in the radial direction, which is perpendicular to the surface \(S\). Thus, the gravitational potential of a point mass \(M\) at distance \(r\) is \(\Phi = -GM/r\). The solution of a differential equation with a delta function source term is in general known as the Green’s function and the Green’s function of the Laplacian \(\nabla^2 \Phi = \delta(\vec{x})\) is therefore \(\Phi = -1/(4\pi r)\).
Using that the gravitational force is the gradient of the potential from Equation (2.1), we find Newton’s law of gravity: at a three-dimensional position \(\vec{x}\) a distance \(r\) from the point-mass \(m\) located at the origin \begin{equation} \vec{F}(\vec{x}) = -\frac{GMm}{r^2}\,\vec{\hat{x}}\,. \end{equation} Putting the position \(\vec{x}_0\) of the mass \(M\) back in explicitly, this force law becomes \begin{equation} \vec{F}(\vec{x}) = -\frac{GMm}{|\vec{x}-\vec{x}_0|^3}\,(\vec{x}-\vec{x}_0)\,, \end{equation} where we have used the standard simplification that the unit vector along \((\vec{x}-\vec{x}_0)\) is \((\vec{x}-\vec{x}_0)/|\vec{x}-\vec{x}_0|\). Because the force falls off as \(1/r^2\) rather than following something like a fast exponential decline, it is called a long-range force. In particular, if a point mass \(m\) is surrounded by a density of point masses that is uniform, then the amount of mass \(M\) in a shell at distance \(R\) is \(\propto R^2\), which combined with the \(1/R^2\) behavior of the force means that the force acting on \(m\) from shells at different \(R\) is approximately constant. In a constant-density medium, there are many more such shells at large distances than at small distances, and therefore the force is dominated by the total contribution from distant shells, rather than that from a few nearby shells.
A pair of functions (\(\Phi,\rho\)) that solve the Poisson equation \((\nabla^2 \Phi = 4\pi G \rho)\) is known as a potential-density pair. Because the Laplacian is a linear differential operator, we have that a linear combination of solutions to the Poisson equation is itself a solution: for potential-density pairs (\(\Phi_1,\rho_1\)) and (\(\Phi_2,\rho_2\)), the pair (\(\Phi_1+\Phi_2,\rho_1+\rho_2\)) is also a solution. A consequence of this is that the gravitational potential for a set of \(N\) point masses is simply given by the sum of the potentials for the individual point masses: If an object at \(\vec{x}\) with mass \(m\) is at a distance \(d_i = |\vec{x}-\vec{x}_i|\) from point masses \(M_i\) at positions \(\vec{x}_i\), the total gravitational potential is \begin{equation}\label{eq-gravitation-Phi-sum-pointmasses} \Phi(\vec{x}) = -\sum_i \frac{GM_i}{d_i}\,. \end{equation} Similarly, the gravitational force is
\begin{equation} \vec{F}(\vec{x}) = -\sum_i \frac{GM_i m}{d_i^3}\,(\vec{x}-\vec{x_i})\,. \end{equation} Note that this follows directly from the Poisson equation, while deriving it from Newton’s law of universal gravitation would require the additional assumption that forces add up linearly (which is of course baked into the Poisson equation). For a continuous distribution of matter, Equation (2.10) becomes \begin{equation}\label{eq-gravitation-Phi-sum-pointmasses-cont} \Phi(\vec{x}) = -G\,\int \mathrm{d}^3\vec{x}'\,{\rho(\vec{x}') \over |\vec{x}'-\vec{x}|}\,, \end{equation} because we can replace \(\sum_i M_i\cdot\) with \(\int\mathrm{d}^3\vec{x}'\,\rho(\vec{x}')\cdot\).
The mass of galaxies is contained in discrete chunks, whether they be stars, putative dark-matter particles, or the atoms and molecules of the interstellar medium. Even though this matter is discrete, the overall distribution of mass is rather uniform, and the gravitational force even between large chunks like stars is therefore dominated by distant bodies (see the argument above). Therefore, we can approximate the density in a galaxy as a smooth function, rather than as a sum over discrete bodies. From the Poisson equation, this means that the gravitational potential and gravitational force are smooth functions as well: Because the density is a second derivative of the potential, the potential is essentially a double integral of the density and, therefore, much smoother than the density.
Newton’s second law (which will be discussed in more detail in the next chapter), states that mass times acceleration equals force. The mass that appears in this equation is the same mass as that appears in the equation between the gravitational potential and force (Equation 2.1; or in Newton’s law of gravitation if you wish). Therefore, we have that \begin{equation} F = -m\,\nabla \Phi(\vec{x}) = m\,\vec{a}\,, \end{equation} or \begin{equation} -\nabla \Phi(\vec{x}) = \vec{a}\,. \end{equation} The motion of an object in a smooth, external gravitational potential therefore does not depend on its mass. This fails when the field is not external, that is, when the object’s mass has an effect on its surrounding mass distribution, which in turn affects its motion through Newton’s second law. But for many applications of galaxy dynamics, a smooth, external gravitational potential is an excellent approximation. It therefore makes sense to introduce the gravitational field \(\vec{g}(\vec{x})\)—the force per unit mass—as \begin{equation} \vec{F}(\vec{x}) = m\vec{g}(\vec{x})\,. \end{equation} because then in a smooth, external potential we have that \(\vec{a} = \vec{g}\). This is known as the weak equivalence principle or the universality of free fall: all objects fall the same in an external gravitational field, whether they be feathers, stones, or stars. Einstein made this principle the centerpiece of his general theory of relativity.
Because most of the time we do not need to consider an object’s mass to discuss its motion under gravity, we typically deal only with the gravitational field and much of the literature on galactic dynamics uses the terms “gravitational force” and “gravitational field” interchangeably and often uses the force symbol \(\vec{F}\) when really the field \(\vec{g}\) is meant. In this book, I will attempt to correctly use the terms “force” and “field”. Similarly, because of the weak equivalence principle, the gravitational field and the acceleration caused by it are the same, and the terms acceleration and force/field are often also used interchangeably. Obviously we can only do this when the force is that due to gravity.
When working with physical quantities and equations, it is often useful to keep their units in mind. From the universality of free fall, we know that the units of gravitational field are the same as those of acceleration: length over time squared. The gravitational field is the spatial derivative of the gravitational potential. Therefore, a gravitational potential has units of length squared over time squared, or more simply, velocity squared. Conversely, from the Poisson equation, we have that \(G\times\mathrm{density}\) has the same units as the spatial derivative of the gravitational field: units of inverse-time squared. \(G\) itself therefore has units of length cubed over mass over time squared. For galactic systems, \(G\) is most usefully expressed as (using the CODATA 2018 version of the recommended values of the fundamental physical constants; Tiesinga et al. 2021) \begin{equation} G = 4.301 \times 10^{-3}\,\mathrm{pc}\,\left(\mathrm{km/s}\right)^{2}\,M_\odot^{-1} = 4.301\,\mathrm{kpc}\,\left(100\,\mathrm{km/s}\right)^{2}\,\left(10^{10}\,M_\odot\right)^{-1}\,, \end{equation} where \(M_\odot\) is the mass of the Sun. Because gravity is such a weak force, measuring \(G\) precisely is difficult and the relative uncertainty of the current measurement is \(2\times 10^{-5}\). This is a much larger relative uncertainty than those of other physical constants, which have typical relative uncertainties of \(\approx 10^{-10}\). However, essentially all of astrophysics is only sensitive to the combination \(GM_\odot\), which is known to a relative uncertainty of \(\approx 10^{-10}\), because it is measured using the orbits of spacecraft in the solar system. It is worth remembering that whenever you see a quoted measured mass outside of the solar system, what is really determined is \(G\) times that mass.
To sum up: The Poisson equation is the fundamental equation one has to solve to obtain the gravitational force due to any mass distribution. Because of this fundamental relation, we will use the terms mass distribution and (gravitational) potential interchangeably (where “gravitational” is typically implied in this context if it is not mentioned in front of “potential”). The (negative) gradient of the potential gives the gravitational field that gives rise to motion and is therefore the only quantity that has physical significance. Thus, we can add or subtract any constant from the potential without changing the dynamics; whenever possible we shall fix this constant such that the potential equals zero at \(r = \infty\).
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