\section{Preliminaries} 

\subsection{Velocity Dispersion: What? Why? How?}


\begin{frame}\frametitle{Section Overview} 

In order to understand my research, you need to know what stellar velocity dispersion is\ldots

\vspace{0.4cm}

\begin{itemize}
 \item Basic definitions.
 \item Performing measurements.
\end{itemize}


\end{frame}





\subsection*{Definitions \& Timescales}
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\begin{frame}\frametitle{Basic Definitions} 

    In \textbf{collisionless} systems: 

    \begin{itemize}
      \item Particles behave as though they are interacting with a smooth potential.	
      \item Individual particle-particle interactions are less important than interaction with global potential. 
    \end{itemize}

    \pause 

    \vspace{0.25cm}

    \textbf{Dissipationless:} Mechanical energy is conserved -- not converted into internal thermal energy.

    \begin{itemize} 
     \item Stars and dark matter particles are assumed dissipationless in simulations.
     \item Gas is assumed to be dissipational.
    \end{itemize}

\end{frame}

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\begin{frame}\frametitle{Basic Definitions} 
  Let $R$ and $M$ be the characteristic radius and total mass of a stellar system, respectively, then  $$\rho=3M/4\pi R^3,$$

  \vspace{0.25cm}

  \textbf{Dynamical timescale:} Characteristic timescale required for a typical star to move a distance $R$, $$t_{dyn}\sim\frac{1}{\sqrt{G\rho}}$$
  
  \pause
  
  \begin{itemize}
   \item timescale required for virialization $(2T=-V)$.
   \item shorter in high-density, central regions of the system.
  \end{itemize}

\end{frame}

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\begin{frame}\frametitle{Basic Definitions} 

    \textbf{Relaxation timescale:} $$t_{relax}\approx \frac{N}{8\ln N}t_{dyn}\sim\frac{N}{8\ln N\sqrt{G\rho}}$$ 
    \begin{itemize}
      \item timescale during which the system cannot be considered collisionless.
      \item stars lose all ``memory'' of their initial orbits (due to close encounters).
      \item timescale required for equipartition of energy
      \pause
      \item $t_{relax}\gg H_0^{-1}$ for galaxies (but not GCs), thus stars in galaxies are approximately collisionless\ldots
    \end{itemize}

\end{frame}

\subsection*{The Jeans Equation}

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\begin{frame}\frametitle{The Meaning of \raisebox{0.04cm}[0cm][0cm]{$\sigma^2_{ij}$}}

For completely collisionless, dissipationless systems, the evolution is described by the \textbf{Jeans Equation} (see Appendix V for derivation), 

\vspace{0.2cm}


\[ \nu\partfrac{\bar{v}_{j}}{t}+\nu\bar{v}_i\partfrac{\bar{v}_{j}}{x_i}=-\nu\partfrac{\Phi}{x_{j}}-\partfrac{(\nu\sigma_{ij}^{2})}{x_i} \]

Where $\nu$ is the number density of particles (i.e., stars), $\bar{v}_i=\bar{v}_i(\mathbf{x},t)$ is the average velocity of stars in the $i$ direction, and  $$\sigma^2_{ij}\equiv\overline{(v_i-\bar{v}_i)(v_j-\bar{v}_j)}$$

\vspace{0.25cm}

$\sigma^2_{ij}$ is the covariance of $(v_i,v_j)$, AKA the velocity dispersion tensor.

\end{frame}



\begin{frame}\frametitle{The Meaning of \raisebox{0.04cm}[0cm][0cm]{$\sigma^2_{ij}$}}

The Jeans equation is an analog of Euler's momentum conservation equation for inviscid fluid flow.  

\vspace{0.2cm}

Jeans: 
\[ \nu\partfrac{\bar{v}_{j}}{t}+\nu\bar{v}_i\partfrac{\bar{v}_{j}}{x_i}=-\nu\partfrac{\Phi}{x_{j}}-\partfrac{(\nu\sigma_{ij}^{2})}{x_i} \]

Euler:\[ \rho\partfrac{\mathbf{v}}{t}+\rho(\mathbf{v}\cdot\nabla)\mathbf{v}=-\rho\nabla\Phi-\nabla P \]

\pause

Thus $\nu\sigma_{ij}^{2}$ is analogous with pressure; it is a stress tensor.

\end{frame}

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\begin{frame}\frametitle{A Few Comments}

\begin{itemize}
 \item The Jeans equations are useful for making analytic and semi-analytic models of galaxies.
 \item They are helpful when trying to \textit{understand} galaxies qualitatively.
 \item Projections of $\sigma^2_{ij}$ are observable (see next few slides). This allows comparison between analytic models and observations.
\end{itemize}

\pause

\vspace{0.05cm}
However, the derivation assumed...
\vspace{0.05cm}

\begin{itemize}
 \item The system is completely collisionless
 \item The system is completely dissipationless
 \item The number of stars is conserved
\end{itemize}

\vspace{0.03cm}

Detailed simulations are needed to understand real galaxies.


\end{frame}

\subsection*{Measuring \sig}

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\begin{frame}\frametitle{The meaning of \raisebox{0.04cm}[0cm][0cm]{$\sigma_*$}}

The projection of $\sigma^2_{ij}$ along the $\hat n_i$ direction is

\[ \sigma_*^2 = \hat n_j \sigma^2_{ij} \hat n_i \]

$\sigma_*$ is known as the line-of-sight stellar velocity dispersion, \[ \sigma_* = \sqrt{\hat n_j \sigma^2_{ij} \hat n_i } = \sqrt{\left\langle{v_n^2}\right\rangle-\left\langle{v_n}\right\rangle^2}\]

where $v_n$ is the line-of-sight component of the stellar velocity.

\vspace{0.25cm}

The quantity $\sigma_*$ is observable\ldots


\end{frame}

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\begin{frame}\frametitle{Measuring $\sigma_*$ Observationally}

\textbf{Assumptions:}

\begin{itemize}
 \item The spectrum of the galaxy is the sum of the Doppler-shifted spectra of the constituent stars. 
 \item The flux received is proportional to the number of stars emitting light.  
\end{itemize}

\pause

\textbf{Basic Steps:} 

\begin{enumerate}
 \item Collect a spectrum.
 \item Transform spectrum to rest frame (z=0) to obtain $F_\lambda$.
 \item Select stellar absorption lines to study (e.g., Ca triplet MgIb).
 \item Select high S/N template absorption lines for a template star (or stars).
\end{enumerate}

\end{frame}

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\begin{frame}\frametitle{Measuring $\sigma_*$ Observationally}
From here, the details of the various methods differ. All methods are functionally equivalent to\ldots\pause
\begin{enumerate}
 \setcounter{enumi}{4}
 \item Construct a model spectrum $f_\lambda$ from the linear combination of shifted templates $T_{\lambda,v_i}$,
\[ f_\lambda =C_\lambda + \sum_{i} w_i T_{\lambda,v_i}  \]

where $C_\lambda$ is the continuum flux and $w_i$ are weights.\pause
\item Find the set of $w_i$ which minimize the residual \[ \|F_\lambda - f_\lambda \|^2 \]
\end{enumerate}

\end{frame}

\begin{frame}\frametitle{Measuring $\sigma_*$ Observationally}

By assumption, $w_i$ is proportional to the number of stars with velocity $v_i$. Thus $w_i$ yields the line of sight velocity distribution (LOSVD).

\pause

\begin{enumerate}
  \setcounter{enumi}{6}
  \item From the LOSVD, compute $\sigma_*$.  
\end{enumerate}
\vspace{0.25cm}
Note that this is a \textbf{flux-weighted} measurement of $\sigma_*$.

\vspace{0.25cm}

\pause

Galaxy modelers tend to measure $\sigma_*$ from the actual velocities of particles in the simulation,\[ \sigma_* = \sqrt{\left\langle{v_n^2}\right\rangle-\left\langle{v_n}\right\rangle^2}\]

Sometimes the mass-weighted means are calculated.

\end{frame}
