As described in the article on the Placidus system, the mundane position is the intersection of the dividing S-shaped curve passing through the planet with the celestial equator.
Knowing the mundane position is extremely important for calculating primary directions. When their mundane positions coincide, two planets are considered spatially conjunct on the celestial sphere.
In this article, we will derive equations for calculating the mundane position in the Placidus system. We have the coordinates of the planet, the observer's latitude, and the local sidereal time (or RAMC) as initial data.
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What does the mundane position look like in the Placidus system?
According to the definition, the mundane position has the same deviation from the prime meridian on the equator as the planet $P$ on its diurnal/nocturnal semi-arc.
The ratio $R$ of meridian distance to semi-arc is the same for both points
$$ R = \frac{MD_P}{SA_P} = \frac{MD_M}{90°} $$
Here $MD_P$ and $MD_M$ are the upper or lower meridian distance of planet $P$ and its mundane position $M$, and $SA_P$ is its diurnal/nocturnal semi-arc.
Mandane Position Equation
We take the upper meridian distance and diurnal semi-arc for the planet above the horizon and the lower meridian distance and nocturnal semi-arc for the planet below the horizon.
If we substitute eq. (1-3) from the article on RAMC, we can rewrite the same equation in the following form:
Here $AD_P$ is the ascension difference of the point $P$ described by equation (2) of ascension difference.
Finally, the equation for the mundane positions of the planet with equatorial coordinates $(RA, D)$ is following:
where the $\phi$ is the observer's geographical latitude. The converse equation is following:
The mundane positions of the house cusps are calculated quite simply. The oblique ascension of the ascendant coincides with the mundane position of the 1st house. By definition, the Placidus house system is constructed so that starting from this degree, we count 30°, then 60°, then 90°, and so on, and draw the curves of positions through these points. They serve as dividing lines in the Placidus system.
This means that the mundane position of the $n$-th house is determined as follows:
$$ MP_\text{Cusp} = OA_\text{ASC} + 30° (n - 1)\tag{4} $$
