In the previous article, we introduced the concept of sidereal time. The sidereal day begins at 00:00, when 0° Aries culminates, and 0° Cancer rises. Therefore, by knowing the current sidereal hour, one can predict the exact degree of the ecliptic that rises and culminates.
Let's derive formulas for ASC and MC. All we know is local coordinates (geographic latitude) and local sidereal time (expressed in RAMC terms).
ASC Equation
Let's look at the figure below:
We have two right spherical triangles:
- First with the sides $OA + AD$, $ASC$ and angle $\epsilon$ between them
- Second with the sides $D$, $AD$ and angle $\epsilon$
Here we use the following notation:
- $OA$ - obliques ascension for ASC
- $AD$ - ascension difference for ASC
- $\epsilon$ - inclination of the ecliptic
- $\phi$ - geographical latitude of the observer
Let's use our handy equations for spherical triangles.
From (3) we have
$$ \sin(AD + OA) = \frac{\tan D}{\tan\epsilon}\tag{1.a} $$ $$ \sin AD = \frac{\tan D}{\tan(90° - \phi)}\tag{1.b} $$
From (6) we have
Let's divide $(1.a)$ by $(1.b)$ and expand the sine of two angles by ($2$)
We will rewrite $\tan(AD + OA)$ in (1.c) in form:
$$ \frac{\tan AD + \tan OA }{1 - \tan AD \tan OA } $$
By substituting $(1.d)$ in the last equation, we have the formula for the ASC:
$$ \tan ASC = \frac{\sin OA}{\cos\epsilon\cos OA - \tan\phi\sin\epsilon} $$
As we discussed earlier, the $OA_{ASC} = RAMC + 90°$. It means that:
MC Equation
Let's look at the figure below:
We have the right triangle with sides $360° - RAMC$, $360°-MC$, and angle $\epsilon$ between them.
From the ($9$) it follows that:
It gives us the equation for the MC:
$$ \tan MC = \frac{\tan RAMC}{\cos\epsilon}\tag{2} $$
Bottom Line
We have derived equations for ASC and MC for a given sidereal time $t = RAMC / 15$ and a given geographical latitude $\phi$
