Ordinary daily activities are so linked with the Sun's position in the
sky that it is natural that civil timekeeping is based on
the hour angle of the Sun (HAS) instead of the hour angle
of the first point of Aries. This is known as
apparent solar time (AST):
AST = HAS
and is zero when the Sun crosses the observer's meridian.
An apparent solar day is defined as the time interval
passages of the Sun across the observer's meridian.
The time it takes the Sun to return to the same point in the sky each day
is slightly longer than one sidereal day, which is the time it takes
the stars to return to the same point in the sky each night.
This is illustrated in Figure 20.
Solar and sidereal time.
An observer at O1 on the surface of the Earth observes the
Sun and a star on the meridian. Taking the distance of the star to be
effectively infinite compared to the Sun, the observer will next observe
the star to transit
at O2 when the Earth has rotated once on its axis.
However, in the time the Earth has taken to rotate once on its axis, it has
also moved in its orbit around the Sun by 360/365.25=0.99°,
from position E1
to E2. The Earth will thus have to rotate on its axis
until the observer is
at O3 for the Sun to appear again on the meridian.
The apparent solar day is therefore longer than the sidereal day by a time
interval equal to the time it takes the Earth to rotate on its axis by the angle
360°/365.25, i.e. approximately 4 minutes. Given that an apparent solar day
is 24 solar hours in length, this means that a sidereal day is only
23h56m long in solar units and represents the
rotation period of the Earth. Another way of
expressing this is that the Sun appears to move backwards amongst the stars
by 4 minutes (or 1°) each day.
Each type of day (apparent, solar and sidereal) can be divided into hours, minutes
and seconds, but the solar version of each is
24h/23h56m = 1.0028 times or 0.28% longer
than the sidereal equivalent.
A sidereal day has a constant length, apart from small fluctuations in the
Earth's rotation rate. However, because the Earth's orbit around the Sun
is an ellipse, the Earth's orbital speed varies during the year, causing
an apparent solar day to vary in length, i.e. the rate of motion of the Sun on
is not uniform. For this reason, and also because the
ecliptic is inclined to the celestial equator (along which the Sun's
hour angle is measured), the rate of change of the
hour angle of the Sun varies from day to day. Hence the apparent solar
time, which is the time measured by a sundial, is not convenient
for most modern purposes because of its variability. Instead, astronomers
use mean solar time (MST), which is measured in terms of a fictitious
body called the mean sun that moves along the celestial equator with
a uniform motion in right ascension equal to the mean motion of the real Sun
(about 4 minutes of right ascension per day). The time between
successive passages of the mean Sun over the observer's meridian
is then constant. This time interval is called a mean solar day.
If the hour angle of the mean sun is HAMS, we can write:
MST = HAMS +/- 12h.
The 12h correction is put in to make the mean solar time
0h at mean midnight rather than at mean noon
when the Sun crosses the
meridian. The plus sign is used when the HAMS lies between 0h
and 12h and the minus sign when the HAMS lies between
12h and 24h.
To define the relationship between the positions of the real Sun and the mean
sun we have to consider the concept in more detail. The situation is
depicted in Figure 21.
The problem is to
obtain a body related to the real Sun, S, but which moves so that its
right ascension increases at a constant rate. It is solved in two
The Sun is said to be at perigee, R, when it is nearest to
the Earth. This
occurs once per year around January 1st. A fictitious body called the
dynamical mean sun, D, is introduced which starts off from perigee
with the Sun, moves along the ecliptic with the mean angular velocity of the
Sun and consequently returns to perigee at the same time as the Sun.
this dynamical mean sun, moving in the ecliptic, reaches the first
point of Aries, , the
mean sun, M, starts off along the celestial equator with the Sun's
mean angular velocity, returning to
at the same time as the
dynamical mean sun.
The positions of the Sun, the mean sun, the dynamical mean sun and the
perigee on the celestial sphere.
The meridian from the north celestial pole (NCP) through the
Sun meets the celestial equator at B so that the arc
B is the
Sun's right ascension (RAS). The right ascension of the
mean sun, RAMS, is the arc
M. The quantity
BM is referred to as the equation of time, E, and is
defined by the relation:
E = RAMS - RAS.
The equation of time varies from a value of about -14.25m to
about +16.25m throughout the year as the mean sun and the
Sun get ahead of each other, as shown in
The annual variation of the equation of time.
Many sundials carry a table of values of the equation of time at various
dates throughout the year, enabling the observer to deduce mean solar
time from the apparent solar time given by the sundial. An example
is shown in Figure 23.
A photograph of an equatorial sundial, clearly showing the
shadow cast by the gnomon. The equation of time is marked
on the stone slab at the foot of the sundial.
Given the relation LST = RAX + HAX we derived
earlier, we can write
LST = RAS + HAS = RAMS + HAMS
E = HAS - HAMS.
The equation of time is therefore equal to the difference in hour angle
between the real Sun and the mean sun.
©Vik Dhillon, 30th September 2009