Sidereal time

If you are working with astrology for a longer time you will remember the use of sidereal time for the calculation of houses. You needed to look up this sidereal time in tables for a specific geographic latitude and doing so you found he cusps for the houses.

You still need sidereal time to calculate houses but your software will take care of this automatically. But of course it is also possible to calculate it yourself.

Sidereal time is defined by the highest position of the equator for a specific location and a specific time. You can define this position on the equator in degrees but also in hours. A full circle consists of 360 degrees or 24 hours. So an hour matches 15 degrees (360 divided by 24). If you use hours you use the term ‘sidereal time’, if you use degrees, you call it ‘right ascension’.

24 hours in sidereal time is 3 minutes and 56 seconds less than 24 hours clocktime. You see this defined in the doagram. OUr clock is based on the rotation of the earth around its own axis and we measure it using he position of the Sun. That is why the Sun is at the zenith at 12 clock during the whole year. But if the earth rotates exactly 360 degree the Sun also processed about a degree; tha tis why we need to catch up for about 4 minutes.

Image: https://en.wikipedia.org/wiki/Sidereal_time

In the diagram you see three times the earth with the line m (meridian) that indicates the highest point for an observer that is placed at the point where line m touches the surface of the earth. The earth to the left has a Sun at its hightest point, this is indicated with the line m. The earth in the middle has processed 360 degrees right ascension (24 hours sidereal time) but the Sun is not at its highest position, the line m still needs to traverse he orange part (which is exaggerated for clarity). The earth to the right is drawn for about 4 minutes later in time and does show the Sun at its highest position.

Formula

The calculation consists of 3 steps:

  1. You calculate sidereal time for 0:00 UT at Greenwich, so for 0º geographic longitude.
  2. Correction for actual UT: You convert the actual UT, the time since 0:00 hours, into sidereal time and add this to the result.
  3. Correction for geographic longitude. 15º geographic longitude is 1 hour sidereal time and to compensate this we add another correction to the sidereal time.

Step 1

Calculate the mean sidereal time (ST) for Greenwich at 0:00 hours UT.

For input you wil use factor T, for the calculation of T see Factor t and delta T. Please note that you need to calculate factor T for 0:00 hours UT, other values will give wrong results. For the calculation of factor T you do not correct for delta T. The result is in degrees which you convert to hours.

The formula in decimal degrees, ST0 is ST at 0:00 hours UT at Greenwich, T is factor T.

ST0 = 100.46061837 + 36000.770053608.T + 0.000387933.T² – T³/38710000

Step 2

To calculate sidereal time for the time of birth you need to apply the following correction:

Add UT birth * 1.00273790935 to ST0

Step 3

Calculate the difference in ST for the place of birth: define the geographic longitude in decimal degrees, eastern longitude is positive and western longitude is negative.
Divide the result by 15 and add the result algebraically to the result from step 2.

Example calculation

Calculate the sidereal time for November 2, 2016 (Gregorian calendar) at 21:17:30 UT at Enschede, the Netherlands (52º 13′ N and 6º 54′ E)

You need the Julian daynumber (see Julian day and Julian day number) and factor T (see Factor t and delta T) but calculated for a 0:00 UT, so for November 2, 2016, 0:00:00 UT.

The Julian Day number is 2457694.5

Factor T is 0.168364134155

Step 1: calculation for 0:00 uur UT at Greenwich

ST0 = 100.46061837 + 36000.770053608.T + 0.000387933.T² – T³/38710000

Enter values for T and powers of T

ST0 = 100.46061837

+ 36000.770053608 . 0.168364134155

+ 0.000387933 . 0.0283464816681

– 0.00477253084225 / 38710000

Simplify

100.46061837

+ 6061.23847881

+ 1.099653567e-05

– 1.23289352622e-10

The result value for ST0 is: 6161.69910818

These are decimal degrees, you substract 6120 degrees (17 x 360) to get a result between 0 and 360 degrees: 41.69910818

Multiply the result with 15 to get decimal hours: 2.7799405453, this is ST0

Step 2: correction for actual UT

Add the correction for the actual UT:

Convert 21:17:30 to decimal hours, the result is 21.291666666667

21.291666666667 . 1.00273790935 = 21.34996132

Add to STO:

2.77994054533 + 21.34996132 = 24.1299018653

Substract 24 hours to get a result between 0 and 24 hours: 0.1299018653

Step 3, correction for geographic longitude

The eastern longitude is 6º54′, decimal 6.9 degrees. Divide this with 15 to get the hours.

6.9 / 15 = 0.46

Local sidereal time will be:

0.1299018653 + 0.46 = 0.5899018653

In hours, minutes and seconds: 0:35:23.6

References