lowercase are central angles and uppercase are triangle vertices
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Sine rule:
$$ \frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C} $$
Cosine rule:
$$ \cos a= \cos b\cos c+\sin b\sin c\cos A $$
Cotangent 4-part rule:
$$ \cos l_i\cos\theta_i=\cot l_o \sin l_i-\cot \theta_o\sin \theta_i $$
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Spherical excess: $E=A+B+C-180\degree$
Area=$Er^2$ (for $E$ in radians)
There are two very important equations here:
LST is defined as the RA of the zenith.
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Equation 1: GST (LST at Greenwich) can be calculated with this equation
$$ \Delta \text{GST} = \,\frac{\textrm{days elapsed}}{365.2422} \times24^h\,+\,\frac{\mathrm{\Delta GMT}}{23h56m4s}\times 24^h $$
This equation essentially relates solar time to sidereal time, by assuming that the RA of the Sun increases uniformly over the course of a year. Hence, the solar time here is the mean solar time.
You will need to compare GST to a reference GST (on a known date&time such as at midnight 1st Jan) to obtain the GST at an arbitrary time and date.
The reason we have to convert to the Greenwich times (sidereal and solar) is due to the difference between mean time and civil time. At Greenwich both are equal, so it is a useful reference point. We can technically apply Equation 1 to any location (GST → LST, GMT → LMT), but we usually do not know LMT, just civil time. But we can convert local civil time to Greenwich civil time by subtracting the timezone, and then that is equal to GMT.
Equation 2: Calculating LST
$$ \text{LST}=\text{GST}+\frac{\lambda}{360\degree}\times24^h $$
This also works for calculating the difference in mean solar times around the world, since it’s the same angle difference.
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Random mildly important fact: Perihelion occurs on January 4th, aphelion occurs on July 4th
(0,0 at sun during vernal equinox, origin always at Earth)
To convert between ecliptic and equatorial coords, use spherical triangle formed by NEP, NCP and star.