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Chapter 7 the Galaxy: Its Structure and Content

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Chapter 7 the Galaxy: Its Structure and Content Chapter 7 The Galaxy: Its Structure and Content 7.1 Introduction Chapter 1 included a brief overview of our Galaxy, while later chapters discussed important processes affecting the Galaxy and other galaxies. Here we bring these concepts together to develop an understanding of the Galaxy itself. This will include a consideration of each of the components (disc, bulge, halo, etc.) in detail. The Milky Way Galaxy is, as far as we know, a typical disc galaxy. Figure 1.8 was a cartoon to remind you of its different components. The luminous parts are mostly a disc of population I stars and a bulge of older population II stars. We live in the disc, with the Sun at a distance R0 = 8.0 kpc from the centre. There is also gas and dust: the gas is mostly observed as an HI layer which flares at large radii. There are HII regions and clusters of young stars. There is some evidence that there are three or four spiral arms in the disc (the dust makes it hard to tell the number). The bulge is accompanied by a bar, though the dimensions of it are unclear. There are some very old stars (and globular clusters of very old stars) in the stellar halo. But the most massive part is the dark matter halo, which is made of dark matter of unknown composition. That is not all: there are also the small companion galaxies. The best known of these are the the Large and Small Magellanic Clouds (LMC and SMC) which are ≃ 50 kpc away; these are associated with a trail of debris, mostly HI gas, known as the Magellanic Stream. Then there is the Sagittarius Dwarf Galaxy which appears to be merging with the Milky Way now. 7.2 The Mass of the Galaxy The mass of the Galaxy enclosed within different radii can be determined using a variety of methods. Observations of the rotation curve provide measurements out to ≃ 12 − 15 kpc. Measurements of the dynamics of globular clusters can constrain the enclosed mass to a greater distance. However, while there are good estimates of the enclosed mass of the Milky Way within different radii, it is not known where the halo of the Milky Way finally fades out (or even if the size of the halo is a very meaningful concept). So the only way to get at the total mass of the Milky Way is to observe its effect on other galaxies. The 112 simplest but most robust of these comes from an analysis of the mutual dynamics of the Milky Way and M31 (the Great Andromeda Galaxy): it is known as the timing argument. This is discussed in the next section. 7.3 The Mass of the Galaxy from Dynamical Tim- ing Arguments The dynamical timing argument relies on modelling the dynamics of the Galaxy and nearby galaxies. The Local Group contains two substantial spiral galaxies, the Galaxy and M31 (it does also contain one less massive spiral, M33, several irregular galaxies of modest mass, and numerous low mass dwarfs). We shall first consider the mass constraint that can be obtained from the dynamics of M31 and the Galaxy, ignoring the other Local Group galaxies. The observational inputs are (i) M31 is 750 kpc away, and (ii) the Milky Way and M31 are approaching at 121 km s−1. (The transverse velocity of M31, if any, is poorly determined at present.) A simple approximation for their dynamics is to suppose that they started out at the same point moving apart with initial velocities from the Big Bang, and have since turned around because of mutual gravity. This is not strictly true of course, because galaxies had not already formed at the Big Bang; however it is thought that galaxies (at least galaxies like these) formed early in the history of the Universe, so the approximation may be acceptable. Writing l for the distance of M31 from the Galaxy, and M for the combined mass of both systems, the equation of of motion for the reduced Keplerian one-body problem is d2l GM = − . (7.1) dt2 l2 Here we shall count time from the Big Bang, so that t = 0 refers to the Big Bang. The current time and separation are t0 and l0. In considering a Keplerian problem without perturbation we are, of course, as- suming that the gravity from Local Group dwarfs and the cosmological tidal field is negligible; but as there are no other large galaxies within a few Mpc this seems a fair approximation. It is not obvious how to solve this nonlinear equation, but fortunately the solutions are well known and easy to verify. There are actually three solutions, depending on the precise circumstances. One solution applies to the case where the combined mass M is too small to halt the expansion and the two galaxies drift further apart for ever: this is not the case we have here. A second solution applies to the limiting case where the mass is just insufficient to stop the motion apart (so dl/dt → 0 and l → ∞ as t → ∞). The third solution applies to the case where the mass is great enough to halt the drift apart and the galaxies fall back toward each other: it is this case that we have here, where the two galaxies are already falling towards each other. This solution is most conveniently expressed in parametric form, as t = τ0 (η − sin η) , 1 2 3 l = GMτ0 (1 − cos η) . (7.2) Here τ0 is an integration constant. The¡ other¢ integration constant has been eliminated by the boundary condition that the two galaxies (or at least the material from which 113 Figure 7.1: The change in the separation l of the Galaxy and M31 with time t since the Big Bang in the model used for the timing argument. they formed) were at the same position immediately after the Big Bang: i.e. l =0 at t = 0. It is easy to show that these equations for t and l are a solution to Equation 7.1 by differentiating them to get d2l/dt2 and substituting them into Equation 7.1. This can be done using d2l d dl d dl dt −1 d dl dt −1 dt −1 = = . = . dt2 dt dt dη dt dη dη dη dη dη µ ¶ µ ¶ µ ¶ Ã µ ¶ ! µ ¶ To determine the total mass M, we first consider the dimensionless quantity t dl sin η (η − sin η ) 0 = 0 0 0 , (7.3) l dt (1 − cos η )2 µ 0 ¶ µ ¶t0 0 where the subscripts in t0 and so on refer to the current time, as is conventional in cosmology. The quantity on the left-hand side can be calculated directly from obser- vational data. Inserting the observed values of l = 750 kpc and dl = −121 km s−1 0 dt t0 and a plausible value of 14 Gyr for t (the age of the Universe), we get 0 ¡ ¢ sin η0 (η0 − sin η0) 2 = −2.32 . (1 − cos η0) This equation can be solved numerically to give η0 = 4.28. Inserting these values into t0 = τ0(η0 − sin η0) from Equation 7.2, we get the constant τ0 = 2.70 Gyr. Then 2 1/3 2 1/3 l0 = (GMτ0 ) (1 − cos η0) gives (GMτ0 ) , and using the value we found for τ0 gives1 12 M ≃ 4.4 × 10 M⊙ . (7.4) 1 −15 −1 3 −2 It is useful to remember G in useful astrophysical units as 4.98 × 10 M⊙ pc yr . 114 From its luminosity and rotation curve, M31 appears to have approximately twice the mass of the Milky Way, i.e. MM31 ≃ 2MGalaxy. Using M = MM31 + MGalaxy, this 12 implies that the mass of Milky Way exceeds 10 M⊙. Estimates for the mass of the 12 luminous part of the Milky Way range from (0.05 − 0.12) × 10 M⊙, which confirms that the majority of the mass of the Galaxy is unseen (it is dark matter). It should be noted that this analysis predicts that the Galaxy and M31 will collide, and consequently merge, at some time, about 3.0 Gyr in the future. However, it fails to take account of the component of the velocity of M31 tangential to our line of sight. M31 and the Galaxy may have a tangential component, and therefore may have enough angular momentum that they may not actually come together. Figure 7.2: Distances and velocities of six Local Group dwarf galaxies, and predictions for different values of GM/τ0 (by Alan Whiting). The timing argument can be applied not only to the Andromeda Galaxy, but also to Local Group dwarf galaxies (which have much less mass and behave just as tracers). Figure 7.2 shows plots of l against dl/dt for some Local Group dwarfs, along with the predictions of the timing argument for different values of GM/τ0. This uses 1 1 dl GM 3 sin η l = GMτ 2 3 (1 − cos η) and = . (7.5) 0 dt τ 1 − cos η µ 0 ¶ ¡ ¢ This model assumes that the dwarfs have been moving on radial trajectories since the Big Bang. 7.4 Kinematics in the Solar Neighbourhood The Milky Way is a differentially rotating system. The local standard of rest (LSR) is a system located at the Sun and moving with the local circular velocity (which is ≃ 220 km s−1). The Sun has its own peculiar motion of ≃ 13 km s−1 with respect to the LSR. 115 The rotation velocity and its derivative at the solar position are traditionally ex- pressed in terms of Oort’s constants: 1 hv i ∂hv i A ≡ φ − φ at R = R 2 R ∂R 0 µ ¶ 1 hv i ∂hv i B ≡ − φ + φ at R = R (7.6) 2 R ∂R 0 µ ¶ Observations show that A = +14.4 ± 1.2 km s−1 (kpc)−1 and that B = −12.0 ± 2.8 km s−1 (kpc)−1.
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