🥢 Parallax:

$$ \begin{aligned} \tan{\theta}\approx&\,\theta \quad \text{ so:} \\ d&=\frac{b}{\theta} \\ d_p&=\frac{1}{p} \end{aligned} $$

Where $d_p$ is in Parsec and $p$ in arcsec

🥐 Angles:

$$ 1\deg=60\text{ arcmin} \quad ; \quad 1\text{ arcmin}=60\text{ arcsec} $$

The angular extend $\theta$ of an object is equal to its linear size $r$ divided by the distance $d$:

$$ \theta=\frac{r}{d} \quad \quad \text{ where: } \theta\approx\tan\theta $$

🌐 Coordinates:

🏄‍♂️ Declination:

• from -90 to +90 deg

🌅 Right ascension:

• from 0 to 24 hours

• total area of the sky: $4 \pi$ sterad

🔨 Solid angle:

Definition: is an area of an object on the sky in steradian = 1 radian squared

• surface area of a galaxy of diameter $\theta$ : $\Omega=\frac{\pi}{4}\theta^2$

🏸 Parallax and proper motion:

Parallax: change in the position due to the motion of the earth (arcsec)

Proper motion: change in position ($\mu$) due to velocity ($v$) of the star in the plane of the sky (arcsec/year)

• $v\left[\text{km}\cdot\text{s}^{-1}\right]=\mu\left[\text{arsec}\cdot\text{yr}^{-1}\right]4.74r[\text{pc}]$
• $\theta\left[\text{arcsec}\right]=\alpha[\text{sec}]15\cos{\delta}$ ($\delta$ is declination)

Line of sight velocity: movement towards/away from us

💡 Luminosity and flux:

Luminosity($L$): power emitted by a star (W)

Flux($F)$: power received at some distance d using a collecting area of 1 $\text{m}^2$