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Massimo Capaccioli » 15.Galaxy dynamics - Part II

Tensor Virial theorem

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 $x_k$ the equation (7) in the previous lecture:

$\displaystyle{\frac{\partial}{\partial t}( \nu \overline{v}_j)+\frac{\partial}{\partial x_i}(\nu \overline{v_i v_j})+\nu\frac{\partial \Phi}{\partial x_j}=0},$

and integrate over the whole volume:

$\displaystyle{\int x_k \frac{\partial}{\partial t}( \nu \overline{v}_j) d^3x=-\int x_k \frac{\partial}{\partial x_i}(\nu \overline{v_i v_j}) d^3x - \int x_k \nu\frac{\partial \Phi}{\partial x_j} d^3x =0.} \hspace{2cm}(1)$

Hereafter the (number) density of stars $\nu$ will be treated as it were the mass density $\rho ,$ by assuming that all the stars have the same mass $m$ [or, equivalently, that the mass function is uniform; show it] and that $\nu \gg 1.$ Using this assumption, the total stellar mass of the system is:

$M = \displaystyle{ \int \nu d^3 x$ , in units of ${m} .$

Tensor Virial theorem

The first term in the equation (1) does not depend on time, thus it can be written as:

$\displaystyle{ \int x_k \frac{\partial}{\partial t}( \nu \overline{v}_j) d^3x= \frac{d}{d t}\int \nu x_k \overline{v}_j d^3x=\frac{1}{2}\frac{d}{dt}\int \nu (x_k \overline{v}_j+x_j \overline{v}_k)d^3x},$

which can be shown to be the derivative of the moment of inertia tensor:

$\displaystyle{I_{jk}=\int\nu x_j x_k d^3x}. \hspace{2cm}(2)$

In fact, the derivative of Eq. (2) is:

$\displaystyle{ \frac{d I_{jk}}{dt}=\int \frac{\partial \nu}{\partial t} x_j x_kd^3x} .$

Using the continuity Eq. (5) of the previous lecture, it can be written as: $\displaystyle{ \frac{d I_{jk}}{dt}=- \int \frac{\partial \nu\overline{v}_i}{\partial x_i} x_j x_k d^3x=\int\nu \overline{v}_i(\delta_{ij} x_k + \delta_{ik}x_j) d^3x=\int\nu (\overline{v}_j x_k + \overline{v}_k x_j) d^3x} .$

Kinetic Energy Tensor

Using the divergence theorem:

$\displaystyle{ \int_V \left({\bf \bigtriangledown}\cdot {\bf F}\right) dV = \int_S \left( {\bf F}\cdot{\bf n}\right) dS}$ ,

the second term of equation (1) becomes:

$-\int x_k \frac{\partial}{\partial x_i}(\nu \overline{v_i v_j}) d^3x = \int \delta_{ki} \nu \overline{v_i v_j} d^3x = 2 K_{kj}$

where $K$ is the kinetic energy tensor. By re-writing the term $\overline{v_k v_j}=\overline{v}_k \overline{v}_j + \sigma_{ij}^2 ,$ the kinetic energy tensor becomes:

$\displaystyle{ K_{jk}=\int\frac{1}{2}\nu \overline{v}_j \overline{v}_k d^3 x+\int\frac{1}{2}\nu \sigma_{jk}^2 d^3 x },$

thus:

$K_{jk}=T_{jk}+\frac{1}{2}\Pi_{jk},$

where the component $T_{jk}$ of the kinetic energy tensor is associated to the ordered (streaming) motions (motion tensor) and the term $\Pi_{jk}$ to the random motions (dispersion tensor). We note that the trace of the tensor $K_{jk}$ : $\displaystyle{ \sum_{j=k}K_{jk}=\int \frac{1}{2}\nu |\overrightarrow{v}|^2d^3x+\int \frac{1}{2}\nu (\sigma_{11}^2 + \sigma_{22}^2 + \sigma_{33}^2) d^3 x= K },$ is the total kinetic energy.

Potential Energy Tensor

Finally the third term:

$\displaystyle{ -\int x_k \nu\frac{\partial \Phi}{\partial x_j} d^3x=W_{jk}} ,$

is the potential energy tensor. By writing the total potential as: $\displaystyle{ \Phi=-G\int\frac{\nu (\textbf{x}')}{|\textbf{x}-\textbf{x}'|}d^3x'} ,$ one obtains:

$\displaystyle{ W_{jk}=G \int x_j \nu(\textbf{x}) \frac{\partial}{\partial x_k}\left( \frac{\nu(\textbf{x}')}{| \textbf{x}' - \textbf{x}|} \right) d^3x d^3x'}=G \int \nu(\textbf{x}) \nu(\textbf{x}') \frac{x_j (x_k' - x_k)}{| \textbf{x}' - \textbf{x} |^3} d^3x d^3x' \hspace{0.5cm}(3)$

By symmetry, the order of integration in equation (3) can be inverted. By taking half of the sum of the two expressions we obtain:

$\displaystyle{ W_{jk}=-\frac{G}{2}\int \nu(\textbf{x}) \nu(\textbf{x}')\frac{(x'_j-x_j)(x'_k-x_k)}{|\textbf{x}-\textbf{x}'|^3} d^3x' d^3x },$

the trace of which can be shown to be the total potential energy of the system:

$W=\frac{1}{2}\int\nu(\textbf{x}) \Phi(\textbf{x}) d^3x .$

Tensor Virial theorem

We can now summarize the results and write the Tensor Virial theorem as:

$\displaystyle{ \frac{1}{2} \frac{d^2 I_{jk}}{dt^2}=2T_{jk}+\Pi_{jk}+W_{jk}}. \hspace{2cm}(4)$

Under the assumption that the material system is confined and stationary,

$\frac{d I_{jk}^2}{dt^2}=0,$

the trace of the equation (4) becomes:

$2T + \Pi = 2K = -W, ~~~~~~(5)$

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 $\Phi$ which is a harmonic function of order n, the scalar (Clausius) Virial theorem states that: $2\langle K \rangle - n\langle \Phi \rangle = 0 ,$ where the brackets indicate a time averaging over an arbitrarily long interval.]

Virial theorem and spheroidal systems

Let’s now consider a very simple application of the Virial theorem to a spherically symmetric particle system with total mass $M.$

The kinetic energy of the system is $K=(1/2) M \overline{v^2} ,$ where $\overline{v^2}$ is the mean-square velocity of the particles [demonstrate the formula using Lagrange's mean value theorem].

From equation (5) we obtain:

$\displaystyle{ \overline{v^2}=\frac{|W|}{M}\sim\frac{GM}{r_g}}, \hspace{2cm}(6)$

where r_g is some characteristic gravitational radius.

Virial theorem and spheroidal systems

We can however assume that, for a given family of galaxies, the following proportionalities exist:

$r_g=k_R R_e,$

$\overline{v^2}=k_\sigma \sigma_0^2,$

$M=k_{ML}L=k_{ML}k_SI_eR_e^2,$

where we have introduced the total luminosity and the mass-to-light ratio, $L$ and $k_{ML} ,$ the shape parameter of the luminosity profile, $k_S,$ and the effective luminosity, $I_e .$

By squaring equation (6) and replacing the above definitions, one obtains:

$k_\sigma^2\sigma_0^4=G^2k_{ML}^2k_Sk_R^{-2} I_eL,$

$\sigma_0^4=G^2k_{ML}^2k_Sk_R^{-2}k_\sigma^{-2} I_e L = \mbox{const.}\times L.$

if $I_e$ 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).

Rotation of elliptical galaxies

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 $z$ axis and seen edge-on (e.g. along the $x$ axis). It must be:

$W_{xx}=W_{yy}$ and $W_{xy}=0\ \ \ (x\neq y),$

since we assume $(x,y,z)$ as principal axes. The only meaningful components of equation (5) are:

$2T_{xx}+\Pi_{xx}+W_{xx}=0$ and $2T_{zz}+\Pi_{zz}+W_{zz}=0.$

Their ratio is: $\displaystyle{\frac{2T_{xx}+\Pi_{xx}}{2T_{zz}+\Pi_{zz}}=\frac{W_{xx}}{W_{zz}}}, \hspace{2cm}(8)$

where $T_{zz}=0$ , since the rotation is assumed about the $z$ axis and $v_z=0.$ For density distributions which are constant on similar and homocentric spheroids, the ratio $W_{xx} / W_{zz}$ depends only on the axis ratio $c/a$ of the spheroid.

Rotation of elliptical galaxies

Suppose we have an oblate and anisotropic rotator.

$T_{zz}=0$ , $T_{xx} = T_{yy}$ thus:

$\displaystyle{ T_{xx} + T_{yy}= 2 T_{xx} = \int \nu \overline{v_\phi}^2 d^3 x = \frac{1}{2} M v_0^2},$

where $M$ is the total mass and $v_0^2$ the mean rotation squared.

Furthermore we have

$\Pi_{xx}=M \sigma_0^2$

and

$\Pi_{zz}= (1-\delta) M \sigma_0^2,$

where $\sigma_0^2$ is the velocity dispersion (about the mean) and $\delta$ is the anisotropy parameter. Substituting into Eq. (8), we obtain:

$\left( \frac{v_0}{\sigma_0} \right)^2 = 2 (1 -\delta) \frac{W_{xx}}{W_{zz}} -2$

Thus, from the observations of $v_0 / \sigma_0$ and ellipticities ( $W_{xx} / W_{zz}$ depends on flattening) we can infer if a galaxy is rotationally supported or dispersion supported.

Virial theorem and spheroidal systems

Since the intrinsic $\epsilon_a$ and the observed $\epsilon_t$ ellipticities are related by the equation:

$(1-\epsilon_t)^2 = (1-\epsilon_a)^2 \sin^2 i + \cos^2 i,$

we can also write: $\epsilon_t (2-\epsilon_t)=\epsilon_a (1-\epsilon_a)\sin^2 i\hspace{2cm} (11) .$

Combining equation (10) with (11), we obtain the dashed curves in the figure, showing the dependence of ellipticity on the parameter $v_0/\sigma_0 ,$ 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.

Credit: Binney and Tremaine, Galactic Dynamics, Princeton Univ. Press.

Virial theorem and spheroidal systems

The left panel (a) of the figure (after Davies et al. 1983) shows the position, in the $(v/\sigma, \epsilon)$ plane, of elliptical galaxies (dots) and spheroids (crosses) with luminosities $L < 2.5\times 10^{10} L_\odot .$ The same plot is shown in the right panel (b) but for ellipticals with $L > 2.5\times 10^{10} L_\odot.$ 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.

See text. From Davies et al., Ap.J., 266, 41, 1983.

Asymmetric drift

Let us now make some simple applications of the Jeans theorem. There is a difference between the local circular velocity, $v_c$ (the Local Standard of Rest), and the mean rotation velocity, $\overline{v}_{\phi},$ of the stellar population of our Galaxy at any given distance from the center. This velocity difference, $\overline{v}_{a},$ 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 $v_a$ is: $v_a=v_c - \overline{v}_{\phi} \sim \overline{v_R^2}/D,$ where $D\sim 120$ km/s. We assume a stationary ( $\partial / \partial t =0$ ) symmetric disk and use the first of the cylindrical Jeans equations.

$\displaystyle{ \frac{\partial (\rho \overline{v_R^2})}{\partial R}+ \frac{\partial (\rho\overline{v_R v_z})}{\partial z} +\rho \left( \frac{\overline{v_R^2}-\overline{v_\phi^2}}{R} + \frac{\partial\Phi}{\partial R}\right)=0} ,$

where we have used now the symbol $\rho$ for the mass density of stars which is obtained by the number density $\nu$ assuming all the stars having the same mass $m$ . On the equatorial plane ( $z= 0$ ), the equation becomes:

$\displaystyle{ \frac{R}{\rho} \frac{\partial (\rho \overline{v_R^2})}{\partial R}+R \frac{\partial (\overline{v_R v_z})}{\partial z} + \overline{v_R^2}-\overline{v_\phi^2} + R \frac{\partial \Phi}{\partial R} =0},$

having divided for $\rho/R$ and $\partial \rho /\partial z =0$ for symmetry about the equatorial plane.

Asymmetric drift

We define the azimuthal velocity dispersion: $\sigma_\phi^2=\overline{(v_\phi-\overline{v}_\phi)^2}=\overline{v_\phi^2}-\overline{v}_\phi^2 ,$

and the circular velocity: $v_c^2=R(\partial \Phi/\partial R),$ which can be inserted in equation (2) to obtain:

$\sigma_\phi^2- \overline{v_R^2} - \frac{R}{\rho} \frac{\partial (\rho \overline{v_R^2})}{\partial R} - R \frac{\partial(\overline{v_R v_z})}{\partial z} = v_c^2 -\overline{v_\phi}^2 = (v_c -\overline{v_\phi})(v_c +\overline{v_\phi})=v_a(2 v_c - v_a) ,$

where we have used the definition of $v_a$ in equation (1).

Since $v_a \ll v_c$ , we obtain Stromberg’s asymmetric drift equation:

$2 v_a v_c \simeq \overline{v_R^2} \left[ \frac{\sigma_\phi^2}{\overline{v_R^2}}- 1 - \frac{R}{\rho \overline{v_R^2}} \frac{\partial (\rho \overline{v_R^2})}{\partial R} - \frac{R}{\overline{v_R^2}} \frac{\partial (\overline{v_R v_z})}{\partial z}\right]=$ $\overline{v_R^2} \left[ \frac{\sigma_\phi^2}{\overline{v_R^2}}- 1 - \frac{\partial \ln(\rho \overline{v_R^2})}{\partial \ln R} -\frac{R}{\overline{v_R^2}} \frac{\partial (\overline{v_R v_z})}{\partial z}\right]}.$

Asymmetric drift

From observations it is found that $\overline{v_z^2} \propto \rho$ ; furthermore, assuming a constant anisotropy in the $(R,z)$ plane, i.e. $\overline{v_R^2} \propto \overline{v_z^2} ,$ we have:

$\displaystyle{\frac{\partial \ln\rho \overline{v_R^2}}{\partial \ln R}\sim\frac{\partial \ln\rho^2 }{\partial \ln R}=2\frac{\partial \ln\rho }{\partial \ln R }}.$

Thus:

$\displaystyle{ 2 v_a v_c\sim\overline{v_R^2} \left[ \frac{\sigma_\phi^2}{\overline{v_R^2}}- 1 - 2\frac{\partial\ln\rho }{\partial \ln R} - \frac{R}{\overline{v_R^2}} \frac{\partial (\overline{v_R v_z})}{\partial z}\right] }.$

The other derivative is more problematic and depends on the orientation of the velocity ellipsoid. We can mention the two most extreme cases.

The first one is when the velocity ellipsoid is aligned with the cylindrical axes $(R,z) ,$ thus $\partial (\overline{v_R v_z})/\partial z =0 .$
A second one is when the velocity ellipsoid is aligned with the spherical axes $(r,\theta),$ thus we have: $\overline{v_R v_z}\sim\pm(\overline{v_R^2}-\overline{v_z^2})(z/R) ,$ with the sign covering all the possible orientation around the equatorial plane.

Asymmetric drift

If we take the average of the two extreme case above, with a little of algebra we realize that:

$\displaystyle{ \frac{R}{\overline{v_R^2}}\frac{\partial (\overline{v_R v_z})}{\partial z}\sim \frac{1}{2} \frac{\overline{v_z^2}}{\overline{v_R^2}} \pm \frac{1}{2} \left( \frac{\overline{v_z^2}}{\overline{v_R^2}}-1 \right)} .$

Then:

$\displaystyle{ 2 v_a v_c\sim\overline{v_R^2} \left[ \frac{\sigma_\phi^2}{\overline{v_R^2}}- 1 - 2\frac{\partial \ln\rho }{\partial \ln R} -\frac{1}{2}\frac{\overline{v_z^2}}{\overline{v_R^2}}\pm\frac{1}{2} \left( \frac{\overline{v_z^2}}{\overline{v_R^2}}-1 \right)\right]}.\hspace{2cm}(3) .$

In Eq. (3) we can use some observational numbers in the neighborhood of the Sun: $R_0/R_d=2.4$ and $v_c=220\ \it km/s$ , to obtain $v_a\sim\overline{v_R^2}/(110\pm7\ km/s) .$

Mass density in the Solar region

Using the third cylindrical Jeans equation it is:

$\displaystyle{ \frac{\partial(\rho\overline{v_z})}{\partial t}+\frac{\partial(\rho\overline{v_R v_z})}{\partial R} +\frac{\partial (\rho \overline{v_z^2})}{\partial z}+\rho\left(\frac{\overline{v_R v_z}}{R}+\frac{\partial\Phi}{\partial z}\right)=0},$

where, again, $\partial / \partial t=0$ 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 $\sim (\overline{v_R^2} - \overline{v_z^2}) z/(R R_d),$ which is a factor $z^2/(R R_d)$ smaller than the third and fifth terms.
Hence, the equation above can be rewritten as:
$\displaystyle{ \frac{1}{\rho}\frac{\partial (\rho \overline{v_z^2})}{\partial z}=- \frac{\partial\Phi}{\partial z}}, \hspace{2cm}(4)$
For a thin disk, the Poisson equation in the proximity of the plane becomes:
$\displaystyle{ \frac{\partial^2\Phi}{\partial z^2}=4 \pi G \rho} ,$
which can be inserted in equation (4) to obtain:
$\displaystyle{ \frac{\partial }{\partial z}\left[ \frac{1}{\rho} \frac{\partial (\rho \overline{v_z^2})}{\partial z} \right]=-4 \pi G \rho}. \hspace{2cm}(5)$

Mass density in the Solar region

If we can measure the density $\rho$ as a function of the height $z$ and the mean-square vertical velocity $\overline{v_z^2}$ 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 $\rho_0=\rho(R_0, z=0) .$ He obtained $\rho_0\sim 0.15 M_\odot/pc^3 ,$ 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.

The mass of spherical systems

We want to use the spherical Jeans equation, re-written in the form:

$\displaystyle{ M(r) =-\frac{\overline{v_r^2} r}{G}\left(\frac{d\ln \rho}{d\ln r}+\frac{d\ln \overline{v_r^2}}{d\ln r}+2\beta\right)}, \hspace{2cm}(6)$

to determine the mass distribution of spherical systems. In this equation all the spatial quantities $(\rho, M, r)$ 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.

Deprojecting the mass (light) distribution

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 $j(r)$ is the spatial light density distribution of the galaxy, then, from the figure, we have that:

$\displaystyle{ I(R)=\int_{-\infty}^{\infty}j(r)ds=2\int_{R}^{\infty}\frac{j(r)rdr}{\sqrt{r^2-R^2}} }, \hspace{2cm}(7)$

where $R$ is the projected radius on the sky, and $s$ 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) $I(R)$ :

$\displaystyle{ j(r)= -\frac{1}{\pi} \int_{r}^{\infty} \frac{dI(R')}{dR'} \left|_{R'=R} \frac{dR}{\sqrt{R^{2} - r^{2}}}\right. }$ .

Geometry involved in the projection along the line-of-sight for the spherical systems.

Projection of the velocity ellipsoid

Assuming no rotation, we compute the projection of the velocity ellipsoid along the line-of-sight:

$\displaystyle{ \sigma_{los} ^2 (R) = \frac{2}{I(R)} \int_R ^{\infty} \overline{(-v_r \sin \phi + v_\phi \cos\phi)^2 } \frac{\rho \, r}{\sqrt{r^2 - R^2}} dr=}$

$~~ \displaystyle{ = \frac{2}{I(R)} \int_R ^{\infty} (\overline{v_r^2} \sin ^2 \phi + \overline{v_\phi^2} \cos ^2 \phi ) \frac{\rho \, r}{\sqrt{r^2 - R^2}} dr=}$

$= \frac{2}{I( R)} \int_R^{\infty} \left( 1- \beta \frac{R^2}{r^2} \right) \frac{\overline{v_r^2} \rho r}{\sqrt{r^2 - R^2}}dr \hspace{2cm}(8)$

where we recall the definition of the anisotropy parameter: $\displaystyle{ \beta= 1- \frac{\overline{v_\phi^2}}{\overline{v_r^2}}} .$

Note that, due to the geometry, the two azimuthal angles $(\theta, \phi)$ are interchangeable.

The two azimuthal angles ? and ? are interchangeable.

Project of the velocity ellipsoid

In the case of isotropic systems: $\beta=0,$ and eq. (8) becomes:

$\displaystyle{\sigma_{los} ^2 (R)=\frac{2}{I(R)} \int_R ^{\infty} \frac{\rho \,r\overline{v_r^2}}{\sqrt{r^2 - R^2}} dr }.$

In this case we may also write the equation for the de-projected velocity dispersion:

$\displaystyle{ j(r)\sigma_{r}^{2}(r)=-\frac{1}{\pi} \int_{r}^{\infty} \frac{d(I\sigma_{P}^{2})}{dR'} \left|_{R'=R} \frac{dR}{\sqrt{R^{2}-r^{2}}}\right. } .$