Equation for ASC & MC

ASC is a shortened term for the word ascendant, or rising degree of the zodiac circle. MC stands for Мedium Сoeli, the midheaven - the culminating degree of the zodiac circle. These two points form the horoscope’s so-called angles . In this article, we will derive formulas for calculating the rising and culminating degrees of the zodiac.

We will start from the assumption that all we know is the local geographic coordinates (more specifically, latitude) and the local sidereal time.

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.

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

$$\tan ASC = \frac{\tan(AD + OA)}{\cos\epsilon}\tag{1.c}$$

Let’s divide $(1.a)$ by $(1.b)$ and expand the sine of two angles by ($2$)

$$\tan AD = \frac{\sin OA \tan\epsilon\tan\phi}{1-\cos OA\tan\epsilon\tan\phi}\tag{1.d}$$

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:

$$\tan ASC = \frac{-\cos RAMC} {\cos\epsilon\sin RAMC + \tan\phi\sin\epsilon}\tag{2}$$

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:

$$\tan(360° - MC) = \cos\epsilon \tan(360° - RAMC)$$

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 $\phi$

$$\begin{align*} &\tan ASC = \frac{-\cos RAMC} {\cos\epsilon\sin RAMC + \tan\phi\sin\epsilon}\tag{1} \\\ &\tan MC = \frac{\tan RAMC}{\cos\epsilon}\tag{2} \end{align*}$$