# Astrodynamics/Time

Time is a fundamental quantity in physics, and nearly all of our equations deal either directly or indirectly with the passage of time.

## Solar Day

A solar day is the time it takes for the Earth to make one complete rotation about its axis with respect to the sun. With respect to inertial space, a solar day equates to approximately 360.986° of rotation. Because of the Earth's slightly elliptical orbit and the obliquity of its rotation axis with respect to its orbital plane, the time between successive transits of the sun are not identical. The sun occasionally lags or leads the mean value of its position in the sky. Thus two types of solar time exist: apparent and mean. Apparent solar time refers to the absolute position of the sun in the sky. Mean solar time refers to the position of the mean sun, which divides the day into 24 hours.

## Sidereal Day

An illustration of the difference between the solar day and the sidereal day. Right: The sun and a distant star initially transit the local meridian at the same time. Center: after the Earth has rotated 360° in inertial space, the star transits again, but because of the Earth's motion around the sun, the sun appears to have moved with respect to the background stars. Left: 24 hours have passed since the first transit and the sun once again crosses the meridian.

The sidereal day is the length of time it takes the earth to make one complete rotation with respect to the stars. A sidereal day is slightly shorter than a solar day: 23h56m04s.09054 of mean solar time, or 0.9972695664 mean solar days.

To complicate matters slightly, sidereal time is also expressed in hours, minutes and seconds. Thus one solar day is equal to 24h03m56s.55536 of sidereal time or 1.0027379093 sidereal days. Sidereal time can best be understood as an angular measure of the Earth's orientation. Greenwich Mean Sidereal Time (GMST) is the sidereal time at the prime meridian and is essentially the angle between the vernal equinox direction and the vector pointing from the Earth's center to the crossing of the prime meridian and the equator.

## Universal Time

Universal Time (UTC) is basically the mean solar time at the prime meridian. It is based solely on the rotation of the Earth with respect to background stars.

It is related to International Atomic Time (TAI), which is a high precision time base based on a weighted average of atomic clocks at national laboratories around the world. Because the rotation of the Earth is non-uniform, periodically leap seconds must be added or subtracted to convert to UTC. Because the rotation of the Earth is slowing gradually with time due to tidal effects from the moon, leap seconds have only ever been added. Since June 30, 2012, TAI is exactly 35 seconds ahead of UTC.

## Converting from Universal to Sidereal

Sidereal time can be approximated from the current date and universal time to within 0.1 seconds by the following equation[1]:

${\displaystyle GMST=6.697374558+0.06570982441908D_{0}+1.00273790935H+0.000026T^{2}}$

Where

${\displaystyle D_{0}=JD_{0}-2451545.0}$
${\displaystyle D=JD-2451545.0}$
${\displaystyle T=D/36525}$

The H term is the current universal time. JD is the current Julian date and JD0 is the exact Julian date of the previous midnight, this value will end with .5. Generally, the final term can be omitted as it accounts for variations that occur over the course of centuries.

To use this equation, one must be familiar with the concept of Julian dates. The concept was introduced in the 16th century as a way to keep track of astronomical events without needing to deal with complex calculations relating to calendar dates. The Julian date is a running count of days elapsed since noon on January 1, 4713BC. Fractions of the day are expressed as a decimal value. Conversion algorithms exist for converting calendar date to Julian date, but they can be quite complicated and will not be discussed here. Below is a reference chart of Julian dates at noon on January 1st of several years[2]. The Julian day number can be calculated by adding the day number of the year and subtracting 1. Please note that Julian dates are integer values at noon, NOT at midnight

Year Julian date
2010 2455198.0
2011 2455563.0
2012 2455928.0
2013 2456294.0
2014 2456659.0
2015 2457024.0
2016 2457389.0