Stars | Notion

Parameter range of stars:

mass: $0.08-\sim120\,\text{M}_\odot$

Radii: $0.01-\sim500\,\text{R}_\odot$

Temp: $300-200,000\,\text{K}$

Luminosity: $10^{-4}-10^6\,\text{L}_\odot$

HR-diagram

Untitled

$X$: temperature of color
$Y$: luminosity or absolute magnitude

Spectral classification scheme in order of temperature:

Class	Spectrum	Colour	Temperature
O	ionized and neutral helium, weakened hydrogen	Bluish	above $31000$ K
B	neutral helium, stronger hydrogen	Blue-white	$9750-31000$ K
A	strong hydrogen, ionized metals	White	$7100-9750$ K
F	weaker hydrogen, ionized metals	Yellowish-white	$5950-7100$ K
G	still weaker hydrogen, ionized and neutral metals	Yellowish	$5250-5950$ K
K	weak hydrogen, neutral metals	Orange	$3800-5250$ K
M	little or no hydrogen, neutral metals, molecules	Reddish	$2200-3800$ K

Memory technique: Oh, Be A Fine Girl Kiss Me

Hydrogen lines

$$ E_n\propto-\frac{1}{n^2} $$

Lowest energy state ($n=1$)
Lowest energy state of hydrogen: $E=-13.6$ eV

The wavelength of the hydrogen line transitions are given by:

$$ \frac{1}{\lambda}=R\left(\frac{1}{n^2_1}-\frac{1}{n^2_2}\right) $$

$n_1$ and $n_2$ are the quantum levels
$R$ is the Rydberg constant

Series names

Transitions to $n_1=1$ are called Lyman series

$n_1=2$ The Balmer series

$\Delta n =1$ are indicated with $\alpha$

$\Delta n=2$ are indicated with $\beta$ etc…

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5a2e0a65-8d23-422e-99e1-3adf844dd3ee/dynamic_time_scale.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5a2e0a65-8d23-422e-99e1-3adf844dd3ee/dynamic_time_scale.png" width="40px" /> The dynamic time scale: The time a star would take to collapse into itself if there was only the gravitational force acting on it

$$ t_{\text{dyn}}\approx\sqrt{\frac{1}{G\bar{\rho}}} $$

$\bar{\rho}$ is the density of the star
For the sun this is around $1000$ sec </aside>

Newton’s shell theorem

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/d671af2c-a118-44e2-b431-157be87d1d22/Hydrostatic.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/d671af2c-a118-44e2-b431-157be87d1d22/Hydrostatic.png" width="40px" /> Hydrostatic equilibrium: Stars must satisfy the following equation to a very high accuracy otherwise they would be unstable, in other words the pressure gradient needs to equal the gravitational force

$$ \frac{\text{d}P}{\text{d}r}=-\frac{Gm(r)\rho(r)}{r^2} $$

$\rho(r)$ is the density at a radius $r$
$m(r)$ is the mass which is inside a radius $r$
$P$ is the pressure gradient </aside>

Ideal gas law (can be applied to stars)

$$ \begin{aligned}P&=nkT\\\text{or: }\quad PV&=nRT\end{aligned} $$