2.3. Circular velocity and dynamical time¶
Next we introduce some of the most basic properties of mass distributions, those of circular velocity and dynamical time. We define these here based on the properties of circular orbits and they therefore require equations for the motion of a body under the influence of gravity. We discuss these equations in far more detail in Chapters 3.1 and 4. All we need for the purposes of this discussion is that for a circular orbit, the centripetal acceleration \(a_r=-v^2/r\) is balanced by the gravitational field. Thus, for a spherical mass distribution using Equation (2.22) we have \begin{equation}\label{eq-vcirc} v^2 = -r\,a_r(r) = -r\,g_r(r) = \frac{G\,M(< r)}{r}\,. \end{equation} This velocity is the circular velocity \(v_c\) at radius \(r\). This equation directly shows that the circular velocity at radius \(r\) is a direct measure of the total mass contained within that radius and measuring the circular velocity as a function of \(r\) therefore directly measures the mass distribution of a spherical distribution. We will see in Chapter 7 that even for very flattened systems, like disk galaxies, the circular velocity is of a similar magnitude as if the flattened mass were distributed spherically. So the circular velocity measures the mass distribution well even for such systems. For reference, the Sun orbits the center of the Milky Way at a distance of \(R_0 \approx 8\,\mathrm{kpc}\) where the circular velocity is \(v_c \approx 220\,\mathrm{km\,s}^{-1}\). Assuming that the mass within the solar orbit is distributed spherically (a rather poor assumption), these values give an enclosed mass of \begin{equation}\label{eq-mass-from-vc-at-sun} M(r< 8\,\mathrm{kpc}) \approx 9\times 10^{10}\,M_\odot\,. \end{equation} By comparing to the mass within 8 kpc in a sophisticated mass model for the Milky Way that is shown in Figure 1.14, you can see that this estimate is actually quite accurate!
We can immediately use the relation between the circular velocity and the mass distribution to understand how we know of the presence of dark matter in galaxies (a preview of what we will discuss in detail in Chapter 8). Interstellar gas orbits the centers of galaxies on approximately circular orbits and the velocity of the gas can be measured through Doppler shifts of radio emission lines (like the 21 cm line). This allowed researchers in the 1970s and 1980s to measure the rotation curves—the circular velocity \(v_c(r)\) as a function of \(r\)—at distances \(r\) where few stars and little gas are observed to be present. What they found was that \(v_c\) is almost constant with \(r\). Re-arranging Equation (2.24) above shows that this implies that the mass distribution behaves as \begin{equation}\label{eq-mass-vc} M(< r) = \frac{v_c^2\,r}{G} \propto r\,; \end{equation} the cumulative mass increases linearly with \(r\). Given that little stellar or gaseous material is observed, this implied the existence of a large amount of dark matter in galaxies.
We can define the crossing time or dynamical time as the time necessary to cross the galaxy, \(t_\mathrm{dyn} \approx R /v\). More formally, we may define the dynamical time at \(r\) as the period of a circular orbit at \(r\), the time it takes to orbit the galaxy (or other stellar system) once \begin{equation}\label{eq-tdyn} t_\mathrm{dyn} = \frac{2\pi\,r}{v_c}\,. \end{equation} Note that this is not the only possible definition, but all other definitions would involve \(r/v_c\), perhaps with a different pre-factor (see the discussion of the homogeneous sphere in Section 2.4.2 below for more justification for this). Using the relation between \(v_c\) and \(M(<r)\) we can express the dynamical time in terms of the average density \begin{align}\label{eq-tdyn-dens} t_\mathrm{dyn} & = 2\pi\,r\,\sqrt{\frac{r}{G\,M(< r)}} = \sqrt{\frac{3\pi}{G\,\bar{\rho}}}\,, \end{align} where \(\bar{\rho}\) is the average density within radius \(r\): \(\bar{\rho} = M(<r)/(4\pi r^3/3)\). Thus we have that \(t_\mathrm{dyn} \propto (G\bar{\rho})^{-1/2}\). This formula provides a simple and general way to estimate the dynamical time of any mass distribution, spherical or otherwise. Systems of high average density, such as the centers of galaxies or the centers of globular clusters, have short dynamical times, while regions of low average density, such as the outskirts of the halos of galaxies, have long dynamical time scales.
For galaxies, the dynamical times are typically hundreds of Myr to Gyr. For example, using the distance and velocity of the Sun within the Milky Way quoted above, we get that the dynamical time at the Sun is \begin{equation} t_{\mathrm{dyn},0} = {2\pi\times 8\,\mathrm{kpc} \over 220\,\mathrm{km\,s}^{-1}} \approx 225\,\mathrm{Myr}\,. \end{equation} Because galactic rotation curves are close to flat, the dynamical time scales approximately as the distance to the center and, thus, it drops to tens of Myr in the inner kpc and increases to 1 Gyr at \(r \gtrsim 50\) kpc. Galaxies are about 10 Gyr old today, which depending on where you are in the galaxy corresponds to a few dynamical times (in the outskirts) to a few thousand dynamical times (in the innermost regions), with the solar neighborhood being about 40 dynamical times old. Compared to, for example, the \(\approx 4.6\times 10^9\) times that the Earth has gone around the Sun, galaxies are dynamically young.
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