> For the complete documentation index, see [llms.txt](https://slowdiveptg.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://slowdiveptg.gitbook.io/notes/galactic-dynamics/week1.md).

# Week1

## Measure the universe

### From Kepler to Newton

**Kepler III**

$$
P^2\propto a^3
$$

where $a$ is the semi-major axis

**Acceleration of circular motion**

$$
f=\frac{v^2}{r}
$$

By assuming circular orbits of planets, we have

$$
f= v^2r^{-1}\propto \frac{r}{P^2}\propto\frac{r}{r^3}=r^{-2}
$$

which is consistent with **Newton II**

$$
F=\frac{Gm\_1m\_2}{r^2}
$$

### Various astrophysical objects

Using the tool

$$
\frac{v^2}{r}=\frac{GM}{r^2}\Rightarrow M=\frac{v^2r}{G}
$$

we can measure

#### The sun

For the first time we are able to measure the **solar mass**

$$
P=2\pi\sqrt{\frac{a^3}{GM\_{\odot}}}\Rightarrow M\_{\odot}=\frac{4\pi^2a^3}{GP^2}=\frac{4\pi^2\times(1.5\times10^8\text{ km})^3}{G\times(1\text{ yr})^2}=2\times10^{33}\text{ g}
$$

#### Star cluster

$$
R\sim1\text{ pc},\ \sigma\sim 20\text{ km/s}\Rightarrow M\sim10^5M\_{\odot},\ \rho\sim10^5\ M\_{\odot}\text{/pc}^{-3}
$$

Dynamic mass $\sim$ Visible mass

#### Galaxy

$$
R\sim10\text{ kpc},\ \sigma\sim 200\text{ km/s}\Rightarrow M\sim10^{11}M\_{\odot},\ \rho\sim10^{-1}\ M\_{\odot}\text{/pc}^{-3}
$$

**Discovery of dark matter** - Dynamic mass $>$ Visible mass

#### SMBH (Sgr A\*)

$$
R\sim0.01\text{ pc},\ \sigma\sim 500\text{ km/s}\Rightarrow M\sim2\times10^{6}M\_{\odot},\ \rho\sim2\times10^{12}\ M\_{\odot}\text{/pc}^{-3}
$$

In such high mass density, the average distance between two stars (solar mass) is

$$
D\sim\frac{1}{2^{1/3}}\times10^{-4}\text{ pc}\sim20\text{ AU}
$$

#### Cosmology

Typical acceleration

Estimation using **hubble timescale** and **speed of light**

$$
a\sim\frac{v^2}{r}\sim\frac{c^2}{10\text{ Gyr}\cdot c}\sim10^{-7}\text{cm/s/yr}
$$

## Two-body problem

Assuming two point mass $m\_1$ and $m\_2$ rotating around the common CoM

$$
m\_1\ddot{\vec{r}}\_1=\frac{Gm\_1m\_2}{|\vec r\_2-\vec r\_1|^3}\left(\vec r\_2-\vec r\_1\right)\\
m\_2\ddot{\vec{r}}\_2=-\frac{Gm\_1m\_2}{|\vec r\_2-\vec r\_1|^3}\left(\vec r\_2-\vec r\_1\right)
$$

Let $\vec r=\vec r\_2-\vec r\_1$, we have

$$
m\_1\ddot{\vec{r}}\_1=\frac{Gm\_1m\_2}{r^3}\vec r,\quad
m\_2\ddot{\vec{r}}\_2=-\frac{Gm\_1m\_2}{r^3}\vec r
$$

**Trajectory**

In an **effective one-body problem**, we consider only the evolution of $\vec r$

$$
\ddot{\vec r}=-\frac{G(m\_1+m\_2)}{r^3}\vec r\equiv-\frac{Gm}{r^3}\vec r
$$

* Test mass moves around $m\_1+m\_2$

**Energy**

$$
\begin{align\*}
E&=\frac{1}{2}m\_1\dot{\vec r}\_1^2+\frac{1}{2}m\_2\dot{\vec r}\_2^2-\frac{Gm\_1m\_2}{r}\\
&=\frac{1}{2}\left\[m\_1\left(\frac{m\_2}{m\_1+m\_2}\right)^2+m\_2\left(\frac{m\_1}{m\_1+m\_2}\right)^2\right]\dot{\vec r}^2-\frac{Gm\_1m\_2}{r}\\
&=\frac{1}{2}\frac{m\_1m\_2}{m\_1+m\_2}\dot{\vec r}^2-\frac{Gm\_1m\_2}{r}\\
&=\frac{m\_1m\_2}{m\_1+m\_2}\left\[\frac{1}{2}\dot{\vec r}^2-\frac{G(m\_1+m\_2)}{r}\right]
\end{align\*}
$$

where

$$
\mu=\frac{m\_1m\_2}{m\_1+m\_2}
$$

is the **reduced mass**, and

$$
\frac{1}{2}\dot{\vec r}^2-\frac{G(m\_1+m\_2)}{r}
$$

is known as the **specific energy** in the effective one-body system

**Angular momentum**

$$
\begin{align\*}
\vec J&=m\_1\vec r\_1\times\dot{\vec r}\_1+m\_2\vec r\_2\times\dot{\vec r}\_2\\
&=\left\[m\_1\left(\frac{m\_2}{m\_1+m\_2}\right)^2+m\_2\left(\frac{m\_1}{m\_1+m\_2}\right)^2\right]\vec r\times\dot{\vec r}\\
&=\mu\vec r\times\dot{\vec r}
\end{align\*}
$$

where

$$
\vec r\times\dot{\vec r}
$$

is the **specific angular momentum**