<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/59837002-4c2b-47ff-a970-bcd591515ad4/worm_1fab1.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/59837002-4c2b-47ff-a970-bcd591515ad4/worm_1fab1.png" width="40px" /> Kepler's 3$^\text{rd}$ law:

• For 1 star

$$ \begin{aligned} P^2&=a^3 \\ P^2&=\frac{4\pi^2a^3}{GM} \\

\end{aligned} $$

• For 2 stars

$$ P^2=\frac{4\pi^2a^3}{G(M_1+M_2)} $$

</aside>

Angular momentum

$$ L=mrv\sin (\phi) $$

• Where $\phi$ is the angle between $\vec{r}$ and $\vec{v}$
• if $\vec{r}\perp\vec{v}$ then the equation simplifies: $L=mrv$

Centripetal force

$$ F=\frac{mv^2}{r} $$

Orbital velocity of a star

$$ v_s=\frac{M_p}{M_s}\sqrt{\frac{GM_s}{r_p}} $$

• $M_p$ is the mass of the planet
• $M_s$ is the mass of the star
• $r_p$ is the distance between the star and the planet

Ways to find a planet

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/3f6e3f70-3b1b-41bb-b515-9a1cecacabbe/radial_velocity.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/3f6e3f70-3b1b-41bb-b515-9a1cecacabbe/radial_velocity.png" width="40px" /> Radial velocity method: use the above equation to measure the shift of a star and see if there is a planet

</aside>

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c8f97a92-61b9-4d86-86a4-1e646e558f05/Transit_method.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c8f97a92-61b9-4d86-86a4-1e646e558f05/Transit_method.png" width="40px" /> Transit method: look at the dip in light of a star

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Doppler shift

$$ \frac{\Delta\lambda}{\lambda}=\frac{v}{c} $$

Temperature of a planet

$$ T=\frac{1}{2\sqrt{r}}\sqrt[4]{\frac{fL}{\pi\sigma}} $$

• $L$ is the stellar luminosity