We have derived the Jeans Equations as a 1-th moment of the CBE. The integration over all the velocity has reduced the space of the applicability of the CBE from the full 6-D phase space to the 3-D physical space. We now derive the 1-st moment version of the CBE in the physical space, and obtain the Tensor Virial equation, which relates to some global dynamical properties of stellar systems. Let us start by multiplying by the equation (7) in the previous lecture:
and integrate over the whole volume:
Hereafter the (number) density of stars will be treated as it were the mass density by assuming that all the stars have the same mass [or, equivalently, that the mass function is uniform; show it] and that Using this assumption, the total stellar mass of the system is:
, in units of
The first term in the equation (1) does not depend on time, thus it can be written as:
which can be shown to be the derivative of the moment of inertia tensor:
In fact, the derivative of Eq. (2) is:
Using the continuity Eq. (5) of the previous lecture, it can be written as:
Using the divergence theorem:
the second term of equation (1) becomes:
where is the kinetic energy tensor. By re-writing the term the kinetic energy tensor becomes:
where the component of the kinetic energy tensor is associated to the ordered (streaming) motions (motion tensor) and the term to the random motions (dispersion tensor). We note that the trace of the tensor : is the total kinetic energy.
Finally the third term:
is the potential energy tensor. By writing the total potential as: one obtains:
By symmetry, the order of integration in equation (3) can be inverted. By taking half of the sum of the two expressions we obtain:
the trace of which can be shown to be the total potential energy of the system:
We can now summarize the results and write the Tensor Virial theorem as:
Under the assumption that the material system is confined and stationary,
the trace of the equation (4) becomes:
which is the known Scalar Virial theorem.
In both cases the Virial theorem describes the (time average) global properties of the stellar system and does not apply to any subsystem.
[Show that, for any confined system depending on a potential which is a harmonic function of order n, the scalar (Clausius) Virial theorem states that: where the brackets indicate a time averaging over an arbitrarily long interval.]
Let’s now consider a very simple application of the Virial theorem to a spherically symmetric particle system with total mass
The kinetic energy of the system is where is the mean-square velocity of the particles [demonstrate the formula using Lagrange's mean value theorem].
From equation (5) we obtain:
where rg is some characteristic gravitational radius.
We can however assume that, for a given family of galaxies, the following proportionalities exist:
where we have introduced the total luminosity and the mass-to-light ratio, and the shape parameter of the luminosity profile, and the effective luminosity,
By squaring equation (6) and replacing the above definitions, one obtains:
if is also constant for the family of galaxies. [But is that true?] We shall return to this very important relation, which goes under the names of Faber and Jackson for ellipticals, and of Tully and Fisher for spirals (in this case the central velocity dispersion is replaced with the maximum velocity).
The Tensor Virial theorem allows us to link the kinematical properties of galaxies to their apparent shapes and rotation speeds. Let’s consider a system with rotation about the axis and seen edge-on (e.g. along the axis). It must be:
since we assume as principal axes. The only meaningful components of equation (5) are:
Their ratio is:
where , since the rotation is assumed about the axis and For density distributions which are constant on similar and homocentric spheroids, the ratio depends only on the axis ratio of the spheroid.
Suppose we have an oblate and anisotropic rotator.
where is the total mass and the mean rotation squared.
Furthermore we have
where is the velocity dispersion (about the mean) and is the anisotropy parameter. Substituting into Eq. (8), we obtain:
Thus, from the observations of and ellipticities ( depends on flattening) we can infer if a galaxy is rotationally supported or dispersion supported.
Since the intrinsic and the observed ellipticities are related by the equation:
we can also write:
Combining equation (10) with (11), we obtain the dashed curves in the figure, showing the dependence of ellipticity on the parameter which measures the balance between the ordered and random motions; from Eq. (9).
Solid lines are labeled by the anisotropy parameter value, dashed lines show the variation of model points when the inclination of the z-axis changes from edge-on (i = 90°) to face-on (i = 0°) view. We see that the effect of the inclination is to make all the anisotropic systems to mimic the case of isotropy.
The left panel (a) of the figure (after Davies et al. 1983) shows the position, in the plane, of elliptical galaxies (dots) and spheroids (crosses) with luminosities The same plot is shown in the right panel (b) but for ellipticals with We note that less luminous galaxies are generally isotropic, while more luminous systems fill the region of the plot where inclined anisotropic systems are expected.
Let us now make some simple applications of the Jeans theorem. There is a difference between the local circular velocity, (the Local Standard of Rest), and the mean rotation velocity, of the stellar population of our Galaxy at any given distance from the center. This velocity difference, is called asymmetric drift and is a consequence of the fact that part of the gravitational pull is balanced by the random motions of the stars, rather than only rotation. The empirical expression found for is: where km/s. We assume a stationary () symmetric disk and use the first of the cylindrical Jeans equations.
where we have used now the symbol for the mass density of stars which is obtained by the number density assuming all the stars having the same mass . On the equatorial plane (), the equation becomes:
having divided for and for symmetry about the equatorial plane.
We define the azimuthal velocity dispersion:
and the circular velocity: which can be inserted in equation (2) to obtain:
where we have used the definition of in equation (1).
Since , we obtain Stromberg’s asymmetric drift equation:
From observations it is found that ; furthermore, assuming a constant anisotropy in the plane, i.e. we have:
The other derivative is more problematic and depends on the orientation of the velocity ellipsoid. We can mention the two most extreme cases.
If we take the average of the two extreme case above, with a little of algebra we realize that:
In Eq. (3) we can use some observational numbers in the neighborhood of the Sun: and , to obtain
Using the third cylindrical Jeans equation it is:
where, again, for the stationary assumption, and the second and fourth terms can also be neglected.
We have seen in fact that these are unlikely to be larger than which is a factor smaller than the third and fifth terms.
Hence, the equation above can be rewritten as:
For a thin disk, the Poisson equation in the proximity of the plane becomes:
which can be inserted in equation (4) to obtain:
If we can measure the density as a function of the height and the mean-square vertical velocity of any population of stars in the solar neighborhood, it is possible to derive the local mass density via Eq. (5). Unfortunately the uncertainties in such estimate are large because it relies on a triple differentiation in the star counts. Oort (1932, 1965) was first to make an estimation of He obtained which is still called the Oort limit in his honor. For more accurate estimates see, e.g., Binney and Tremaine, 1987, Galactic Dynamics, Princeton Univ. Press, pp. 200-201.
We want to use the spherical Jeans equation, re-written in the form:
to determine the mass distribution of spherical systems. In this equation all the spatial quantities are written in spatial coordinates, while observations are projected on the plane of the sky. Similarly, the measured velocity moments are only available along the line-of-sight, while in principle Eq. (6) contains the information of the full 3-D velocity space. Thus it must be solved only after having “de-projected” the observed quantities. This procedure has two complications. One is physical, i.e. we need to make assumptions on the internal structure of the orbital distribution (while the spherical geometry is simple to handle and provides also a unique de-projection). The other one is technical, as observations are noisy and the de-projection contains the derivative of the measured quantities, which amplifies the noise [why?], making the result quite uncertain. Let us start by writing the equations which connect the projected observed quantities and their de-projections.
We start by re-computing the projected light density of the system (i.e. the surface brightness), under the implicit presumption that it is proportional to the matter density (mass-to-light ratio constant). If is the spatial light density distribution of the galaxy, then, from the figure, we have that:
where is the projected radius on the sky, and is the coordinate along the line-of-sight. It is apparent that we have used circular symmetry and assumed that there is no internal extinction. This equation has a well know Abell kernel that can be easily inverted to give the deprojected version of the (observed) :
Assuming no rotation, we compute the projection of the velocity ellipsoid along the line-of-sight:
where we recall the definition of the anisotropy parameter:
Note that, due to the geometry, the two azimuthal angles are interchangeable.
In the case of isotropic systems: and eq. (8) becomes:
In this case we may also write the equation for the de-projected velocity dispersion:
8. Spiral arms
10. Scale relations
14. Galaxy dynamics
19. Galaxy clusters