## Musings on Days and Sidereal Time

I like to review periodically some basic concepts of astronomy to get a better perspective on what we do as astrologers. Perhaps the most basic idea of all is the length of a day, which is based on the rotation of the Earth. Usually when we say “day” we are referring to the 24 hours of a mean solar day, which is the average length of time it takes the Sun to appear on the meridian of a location twice in a row. We are accustomed to using the Sun to tell time, and one complete daily cycle of the Sun around the Earth (from our position as observers on this planet) is allotted 24 hours, giving us our 24-hour-clock mean solar day as a unit of measurement.

Just as there is a mean solar day, whose duration is based the appearance of the Sun on the meridian of a location from one day to the next and assigned the value of 24 hours, there is also a mean sidereal day, which is the average length of time it takes a fixed star (hence sidereal) to appear on the meridian of a location twice in a row.

The following image from wikipedia illustrates the difference between a solar day and a sidereal (fixed star based) day: This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.  In the above diagram you can see that the Earth makes a complete rotation in 23h 56m 04s relative to the fixed starts but it needs another 3m 56s of travel on its orbit for the meridian (m) of the observer to align with the Sun.  Covering this additional distance on its orbit adds a little less than one second of “correction” to the sidereal time of the moment when the Sun aligns with the meridian, so that every 24-hour day the ephemeris adds roughly 3m 57s to the previous day’s sidereal time, as you can see in the ephemeris example later in this post.

A sidereal day refers to one complete rotation of the Earth on its axis, as measured against the fixed stars. Each complete rotation of the Earth requires 23 hours 56 minutes 4.0905 seconds (or 23.9344696 hours) of 24-hour-clock time — which is almost 4 minutes shorter than a mean solar day. What happens during those extra 4 minutes?

Wikipedia gives the following explanation: “Earth makes one rotation around its axis in a sidereal day; during that time it moves a short distance (about 1°) along its orbit around the Sun. So after a sidereal day has passed, Earth still needs to rotate slightly more before the Sun reaches local noon according to solar time. A mean solar day is, therefore, nearly 4 minutes longer than a sidereal day” (https://en.wikipedia.org/wiki/Sidereal_time).

In other words, the Earth keeps moving along its orbit, and after about 4 minutes the MC of the observer finally aligns again with the Sun. The implications and underpinnings of the Earth continuing along its orbit around the Sun after having made a complete rotation on its axis are the following:

• A sidereal year (the amount of the required for the Earth to complete one orbit around the Sun with respect to the fixed stars) has a duration of 365.256363004 mean solar days.  A mean sidereal year is longer than a mean tropical year by 20 minutes 24.5 seconds.
• A tropical year (the amount of time needed for the ecliptic longitude of the Sun to increase by 360 degrees, taking into account the precession of the equinoxes) has a duration of 365.242190402 mean solar days.
• Solar time is measured by the apparent diurnal motion of the Sun.
• Local Noon in apparent solar time occurs when the Sun crosses the observer’s meridian.
• A mean solar day is the average length of time between consecutive local solar Noons over the course of a year.
• A mean sidereal day has a duration of 23 hrs 56 min 4.0905 sec of solar 24-hour-clock time (23.9344696 hours of solar time).  This is the amount of time required for the Earth to make one complete rotation on its axis with respect to the fixed stars.
• Both solar time and sidereal time are based on the daily rotation of the Earth about its North and South Poles. The difference is that solar time also takes into account the movement of the Earth on its orbit around the Sun to measure one rotation of the Earth, whereas sidereal (star-based) time measures the Earth’s rotation with reference to the fixed stars. These are simply measure of time from different perspectives. An observer on the Sun of our solar system would measure the Earth’s rotation in solar time. An angel looking down on the entire universe from heaven would measure the rotation of the Earth in sidereal time.
• The Earth makes a complete rotation every 23.9344696 hours of solar time, and the length of the mean tropical year is 365.242190402 solar days. Thus, we can divide the length of the year by the amount of time for each rotation to calculate the number of complete rotations of the Earth per tropical year:  (365.242190402d x 24h) divided by 23.9344696h = 366.2419 complete rotations of the Earth per tropical year.
• A mean solar day therefore has a duration of 24 hrs 3 min 56.56 sec of sidereal time.  [If you look in the ephemeris, the sidereal time listed for each day is approximately 3 min 57 sec of sidereal time later than that of the previous day.]
• The Naibod Key for Primary Directions is a based on the mean daily motion of the Sun, which Naibod calculated to be about 59′ 8″ of arc per day is equivalent to one year of life, according to the calculation: 360 degrees in a complete circle divided by 365.242190402 mean solar days in a tropical year.  [More precisely, 360 degrees divided by 365.242190402 = 59′ 8.33″ of arc.]

Those astrologers who calculate (or have calculated) natal charts by hand know that we must use the sidereal time of birth to identify the MC of the chart.  Astronomers use sidereal time to indicate when (at what time) a celestial object will appear in the sky and celestial latitude above or below the Celestial Equator to indicate where that object will be located. Knowing the sidereal time and latitude of a star, astronomers are able to properly point their telescopes to view it.

Sidereal time is a measure along the Celestial Equator of the arc between the March Equinox (where the path of the Sun crosses the Celestial Equator in March) and the meridian of the observer. Because sidereal time is a measure both of time and of angular distance (an arc), it can be expressed either in hours, minutes and seconds, as on a clock, or in degrees of Right Ascension along the Celestial Equator. By convention 0 Aries is assigned to the March Equinox in the tropical zodiac, and thus 0h 0m 0s of sidereal time corresponds to 0 Aries on the ecliptic.

The reason that astronomers use sidereal time is that at a given location on Earth a celestial object will appear at a particular position in the sky with reference to the Celestial Equator at the same sidereal time every night. Sidereal time differs from our usual 24-hour-clock time, so it is necessary to convert between them, kind of like Americans converting from Dollars to Euros when they travel to Europe. To see how this conversion works, let’s take a look at an ephemeris.  Below is a section of the Astrodienst ephemeris from September of 1945. We see that on 1 Sep 1945, the sidereal time (ST) at 0 hours in Greenwich, UK, is 22h 39m 10s.  Twenty-four hours or one solar day later (2 Sep 1945) the sidereal time (ST) is 22h 43m 7s.  The difference between these two values of ST is 3m 57s.  (During this 24-hour period the Sun has advanced along the ecliptic by 58′ 6″.)

As can be seen in the second column of this ephemeris, every solar day of 24 hours, the sidereal time increases by 3m 57s from the previous day. This difference between solar and sidereal times is sometimes called the “acceleration interval” or the “solar-sidereal correction.” To be more precise, for every hour of solar time that passes on the conventional clock, the sidereal time increases by that solar hour plus an additional 9.8333 seconds (hence the idea of an “acceleration interval”).  This rate is often rounded to 10 seconds of additional sidereal time for every one hour of solar time, which is equivalent to roughly 1 second of additional sidereal time for every 6 minutes of solar time.

For example, we saw previously that each complete rotation of the Earth (one sidereal day) requires 23 hours 56 minutes 4.0905 seconds of 24-hour-clock time. The solar day does not commence until about 4 minutes (or more precisely, 3m 56s) later than the sidereal day.  During the roughly 4 minutes between the sidereal day and the solar day, the acceleration rate indicates that we must add about 0.65 seconds of solar-sidereal correction to the 3m 56s to get the sidereal time at the start of the solar day. Performing the addition, we get the 3m 57s that appears in the ephemeris as the difference in sidereal time between one solar day and the next. 