6.2. The escape velocity near the Sun and the total mass of the Milky Way¶
We already briefly discussed the escape velocity in the solar neighborhood and its implications for the total mass of the Milky Way in Chapter 3.2. In this section, we take a closer look at how the escape velocity is determined from observational data.
To determine the escape velocity, we consider a kinematically-unbiased sample of stars—meaning that the stars in the sample were not selected for or against based on their velocity or proper motion—and look for a high-velocity cut-off. If a high-velocity cut-off exists, a natural interpretation of that cut-off is that it represents the escape velocity. The cut-off is due to stars with higher velocity never returning to the solar neighborhood. Making this measurement is difficult. By their very nature, stars with velocities near the escape velocity are on orbits with large orbital times which visit the solar neighborhood near their pericenters. Due to conservation of angular momentum, stars spend relatively little time close to their pericenters compared to the rest of their orbits. Therefore, these stars do not spend much time near the Sun for us to observe them. Current samples of stars with velocity measurements near the Sun have millions of stars, yet only a few hundred of these have velocities within \(\approx 200\,\mathrm{km\,s}^{-1}\) of the likely escape velocity that could be used to look for a cut-off. Because of the paucity of data, we need to compare the data to parameterized models of the velocity distribution that include the cut-off at \(v_\mathrm{esc}\).
Stars with high velocities near the Sun spend most of their orbits well into the dark-matter halo rather than near the disk. It therefore makes sense to model them as if they are orbiting in a spherical potential (assuming that the halo is spherical) and we can apply the framework from the previous two chapters. The simplest possible assumption is that the distribution function of high-velocity stars is an ergodic distribution function \(f(\mathcal{E})\), similar to the ones that we discussed in Chapter 5.6. As \(\mathcal{E} \rightarrow 0\), ergodic distribution functions with a finite mass approach \(f(\mathcal{E}) \rightarrow 0\) and we can very generally model this approach using a power law of \(\mathcal{E}\). We therefore posit a distribution function of the form \begin{equation} f(\mathcal{E}) = \mathcal{E}^k\,, \end{equation} for some \(k\). It is easy to see that this is the case for the Hernquist distribution function that we discussed in Section 5.6.1 and for the King models in Section 5.6.3. Remembering that \(\mathcal{E} = \left(v_\mathrm{esc}^2-v^2\right)/2\) (Equation 5.56), this becomes a probability distribution \(p(v|\vec{x})\) of the magnitude \(v\) of the velocity at a given position \(\vec{x}\) (in this case, the solar neighborhood, which can be considered a single position for this application): \begin{equation}\label{eq-df-vesc-ergodic} p(v|\vec{x};v_\mathrm{esc},k) \propto f(\mathcal{E}) \propto \left(v_\mathrm{esc}^2-v^2\right)^k\,. \end{equation} A form similar to this was first proposed by Leonard & Tremaine (1990), but dropping the \((v_\mathrm{esc}+v)^k\) factor when decomposing the right-hand side of Equation (6.5) as \((v_\mathrm{esc}+v)^k\,(v_\mathrm{esc}-v)^k\). The argument made in this paragraph was first made by Kochanek (1996). Because proper motions of stars are difficult to measure (although Gaia is dramatically changing this situation!), data samples that can be used to determine the escape velocity have been typically limited to line-of-sight velocities \(v_\mathrm{los}\) and we therefore need to integrate Equation (6.5) over the tangential velocity; this gives \begin{equation}\label{eq-df-vesc-ergodic-vlos} p(v_\mathrm{los}|\vec{x};v_\mathrm{esc},k) \propto \left(v_\mathrm{esc}^2-v_\mathrm{los}^2\right)^{k+1}\,. \end{equation}
The distribution function in Equation (6.6), with parameters \((v_\mathrm{esc},k)\), can be fit to observational data on the line-of-sight velocities \(v_{\mathrm{los},i}\) of a sample of high-velocity stars. Because the velocities of these stars are assumed to be independent draws from \(p(v|\vec{x};v_\mathrm{esc},k)\), the total probability of all observations \(v_{\mathrm{los},i}\) is \begin{equation} p(\{v_{\mathrm{los},i}\} | v_\mathrm{esc},k) \propto \prod_i p(v_{\mathrm{los},i}|\vec{x};v_\mathrm{esc},k)\,. \end{equation} Maximizing this probability over \((v_\mathrm{esc},k)\) gives the maximum-likelihood estimate of \((v_\mathrm{esc},k)\).
This type of analysis was performed using the RAVE data by Smith et al. (2007) and later updated by Piffl et al. (2014). The resulting measurement of \(v_\mathrm{esc}\) is \(v_\mathrm{esc} = 530\pm50\,\mathrm{km\,s}^{-1}\). Smith et al. (2007) also tested that the distribution function model of Equation (6.5) is a good model using realistic simulations of the Milky Way’s high-velocity stars. There is good reason to doubt the form of Equation (6.5): As we already discussed, the orbital times of stars with \(v \approx v_\mathrm{esc}\) are very long and the assumption of a steady-state distribution function is therefore suspect; furthermore, the process that populates the \(v \approx v_\mathrm{esc}\) tail of the velocity distribution is uncertain and it does not have to be the case that the distribution function would not reach zero at a non-zero \(\mathcal{E}\). However, Smith et al. (2007) found that the form in Equation (6.5) is a good enough model for the current purposes at least for the simulations. An update of the Smith et al. (2007) analysis using the Gaia data and using simulations more tailored to the Milky Way by Deason et al. (2019) found \(v_\mathrm{esc} = 530\pm 25\,\mathrm{km\,s}^{-1}\).
The escape velocity is a unique constraint on the mass of the Milky Way because, unlike most dynamical constraints, it is sensitive to the cumulative mass outside of the solar circle. As we discussed in Chapter 3.2, unfortunately we have to assume a density distribution for the mass outside of the solar circle, so any determination of the Milky Way’s mass using the escape velocity at the solar circle depends on the assumed density of matter outside of the solar circle. Assuming the cosmologically-motivated NFW profile, the measurement of \(v_\mathrm{esc} = 530\pm 25\,\mathrm{km\,s}^{-1}\) gives a total halo mass of \(M \approx 1.00^{+0.31}_{-0.24}\times10^{12}\,M_\odot\) (Deason et al. 2019).
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