A sidereal day is a time for a complete rotation of the celestial sphere around its axis. A current portion of the sidereal day is called a sidereal time.
The line from South to North through the zenith is called the Prime Meridian. Any point of the celestial sphere passing through the primary meridian by its daily motion locates strictly between the eastern and western horizons, that is, at its maximum elevation on the way from ascension to descension. This position is called the culmination (a maximum elevation during a daily motion).
Astrologers have agreed to consider the beginning of a sidereal day the moment of culmination of 0° of the celestial equator.
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Connection of Sidereal Time and RAMC
Suppose you are somewhere in the middle of the sidereal day. If at the beginning of a sidereal day, 0° of the equator culminates, then a degree with the coordinate RA (right ascension) will culminate at the moment. This degree is called the Right Ascension of the Midheaven, or Medium Coeli (MC), in Latin. This degree is abbreviated as RAMC.
RAMC.
Since 24 hours of sidereal time corresponds to 360 degrees of a complete revolution of the celestial sphere, then one sidereal hour corresponds to 15 degrees of the celestial equator. So we can easily set the current sidereal time, knowing the current culminating degree:
$$t_{hrs} = \frac{RAMC} {15}$$
Upper/Lower Meridian Distance and Sidereal Time
Each planet has its equatorial coordinate, RA (Right Ascension). The angular distance between RAMS and RA is called the Upper Meridian Distance (UMD).
If we express the same distance in terms of sidereal time between the current position of the planet and its upper culmination, then we get the hourly distance, HD
$$HD_{hrs} = \frac{UMD} {15}$$
UMD.
The figure above shows that the sum of RA and UMD is equal to RAMC:
$$ RA + UMD = RAMC $$
In the most general case
$$\begin{align} UMD &= |RAMC - RA| \\ &= \left| t_{hrs} \times 15 - RA \right| \end{align}\tag{1}$$
Finally,
$$ HD = \left| \frac{RA}{15} - t_{hrs} \right| $$
P.S. If the planet is below the horizon, astrologers use the Lower Meridian Distance (LMD), the planet's distance from the point of its lowest culmination. It is connected with $UMD$ by a simple formula:
$$\begin{align} LMD &= UMD + 180° \\ & = |RAIC - RA_P| \tag{2} \end{align}$$
Here $RAIC$ is the right ascension of Imum Coeli $RAIC = RAMC + 180°$
RAMC and Oblique Ascension of ASC
We already know that the oblique ascension of the zodiacal ASC is the degree of the equator ascending simultaneously with the ASC at a given time.
But we have already discovered that the culminating degree at any point in time is RAMC. Therefore, if we add 90 degrees to the RAMS, we get the currently rising equator's degree. And this is the oblique ascension of the ASC.
Oblique ascension of ASC.
That is:
$$ OA_{ASC} = RAMC + 90° $$
Diurnal/Nocturnal Semi-Arcs and Sidereal Time
A path of any given point from the horizon to the prime meridian is called a diurnal/nocturnal semi-arc.
It is called diurnal semi-arc, DSA for the path beyond the horizon and nocturnal semi-arc, NSA.
DSA.
As it is clear from the figure above, diurnal semi-arc, $DSA$ equals to
$$\begin{align} DSA &= |RA - RAMC| \\ &= |RA - t_{hrs} \times 15| \end{align}$$
For the nocturnal semi-arc, we use the right ascension of Imum Coelum, $RAIC = RAMC + 180°$, instead of $RAMC$:
$$\begin{align} NSA &= |RA - RAIC| \\ &= |RA - 180° - t_{hrs} \times 15| \end{align}$$
From the same figure 4, where $AD$ is negative, it is also clear that
$$ \begin{cases} DSA = 90° + AD \\ NSA = 90° - AD \end{cases}\tag{3} $$
Here $AD$ is the ascension difference with the equation of the AD.
