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