Lecture 7
Time Systems
Time is used as a fundamental dimension in almost every branch of science. When a scientist or layman uses
the terms “hours, minutes or seconds” he is understood to mean units of mean solar time. This is the time
kept by ordinary clocks and in common use in western society and adjusted for locality. To understand the
rotation of the earth in inertial space we will need to talk about another kind of time called “sidereal” time.
Computing the position of a satellite in orbit about a planet or star, whether man made or natural, depends
on the existence of an accurate timeing mechanism used with a time keeping system. Mechanisms for keeping
track of time are ancient; the burning of candles, the dripping of water, the motion of a pendulum, the length
of a day, the phases of the moon. Some more modern devices include the oscillation of an inductive and
capacitive circuit, the piezoelectric resonance of a quartz crystal, and the electronic transition of an atom.
In this lecture we provide the background in time keeping systems needed for space systems engineering that
rely on both historic and modren time keeping mechanisms.
7.1
Solar and Sideral Time
It is the sun more than any other heavenly body that governs our daily activity cycle; so, it is no wonder that
ordinary time is reckoned by the sun. The time between two successive transits of the sun across our local
meridian is called an “apparent” solar day. The Earth has to turn through slightly more than one complete
rotation on its axis relative to the ”fixed” stars during this interval. The reason is that the earth travels
about 1/365th of the way around its orbit during the period of one day. An Appart Solar Day can be divided
into 24 hours with 60 minuits per hour and 60 seconds per minute.
A sidereal day is defined as the time required for the earth to rotate once on its axis relative to the stars.
It is the time between to successive transits of a star across a local meridian. This occurs in about 23h 56m
4s of ordinary solar time. A sideral day is also broken into 24 sideral hours with 60 minutes per hour and 60
secons per minute which leads to the following approximate relationships:
• 1 day of mean solar time = 1.0027379093 days of mean sidereal time = 24h 3m 56.55536s of sidereal
time = 86636.55536 mean sidereal seconds
• 1 day of mean sidereal time = .9972695664 days of solar time = 23h 56m 04.09054s of mean solar time
= 86164.09054 mean solar seconds.
So far, we have really defined only local sidereal time and ”apparent” solar time. Based on the definition
of an apparent solar day illustrated in the figure, no two solar days would be exactly the same length because
the Earth’s axis is not perpendicular to the plane of its orbit and because the earth’s orbit is slightly elliptical.
In early January, when the Earth is near perihelion, it moves farther around its orbit in 1 day than it does
in early July when it is near aphelion. In order to avoid this irregularity in the length of a solar day, a mean
solar day is defined based on the assumption that the Earth is in a circular orbit whose period matches the
actual period of the Earth and that the axis of rotation is perpendicular to the orbital plane. An ordinary
clock which ticks off 24 hours in one mean solar day would show the sun arriving at our local meridian a
little early at certain times of the year and a little late at other times of the year. A mean solar day is used
for ordinary timekeeping which is the average of an apparent solar day over a year.
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LECTURE 7. TIME SYSTEMS
26
In March 1675 the King of England established the Royal Observatory in Greenwich to improve navigation at sea and ’find the so-much desired longitude of places’ while at sea and out of sight of land, by
astronomical means. This goal was inseparable from the accurate measurement of time, for which the Royal
Observatory became generally famous in the 19th century. The internationally adopted prime meridian, 0
degrees longitude, runs through a particular mark at the former site of the Royal Observatory in Greenwich,
England. Mean solar time at the prime meridian is defined as Universal Time (UT), also called Greenwich
Mean Time (GMT) or Zulu Time (Z). Historically, this observatory and the development of an accurate
marine chronometer in the 1750’s, allowed the British seamen to expertly navigate the oceans.
Greenwich Mean Time, which evolved from observations of the mean solar day, has become a somewhat
ambiguous term in modren usage. This is due to the introduction of more percise time keeping methods
and more percise observations of the earth’s motion in space with respect to some 212 extragalactic sources
such as quasars. The term GMT now generally refers to a “Universal Time” that tracks time independently
of one’s location on the Earth. There are a number of variations on the theme of universal time, each
appropriate in its own context:
UT0 is universal time as measured using the daily motions of stars or extragalactic radio sources. It is rarely
used because it is not truly universal as it does not take into account changes in the displacement of
the earth’s poles (”polar motion”), and so different observatories will measure different UT0 intervals,
even though they are looking at the same objects.
UT1 is the principal form of Universal Time in use today. It is UT0 corrected for polar motion, and so
is the same everywhere on earth. It changes slowly over time because of the steady slowdown in the
earth’s rotation rate, and the irregular jitter in the rotation rate due to motions in its fluid interior
which amount to about +/-3 milliseconds per day. UT1 is effectively the same as Greenwich Mean
Time (GMT) as it was before the introduction of atomic clocks.
TAI is the International Atomic Time scale. It is a statistical timescale based on a large number of atomic
clock operating at standards laboratories around the world. The unit interval is the Systeme International (SI) second defined as the duration of 9,192,631,770 cycles of radiation corresponding to the
transition between two hyperfine levels of the ground state of cesium 133. TAI is a free running number
of seconds and TAI-UT1 was approximately 0 on Jan 1, 1958.
UTC UTC is Coordinated Universal Time and is measured using TAI and is the basis of all civil timekeeping
on Earth today. It is kept to within +/-0.9 seconds of UT1 by adding (or potentially removeing) leap
seconds to the TAI base every few years to keep the position of the sun close to the prime meridian at
at noon. All civil atomic clocks and computer time services provide time in UTC.
Some important relations and defintions for solar and siderial time:
• Day (Apparent Solar Day) is appearance of sun over a given meridian
• (Mean Solar Day) is average of the apparent solar days over one year.
• Mean solar time at the prime meridian is defined as Universal Time (UT) which is also called Greenwich
Mean Time (GMT) or Zulu Time (Z).
• Variation in the Earths rotation period over weeks and months makes the Earth a poor time base from
a physics standpoint. Scientists and Engineers have esentially seperated the measure of time from the
motions of the Earth through the use of atomic clocks (TAI). UT (UT1) is the non-uniform time system
that is synchronous with the rotation of the Earth and is the logical continuation of GMT today. UTC
is a uniform time system based on TAI that is occasionally adjusted with leap seconds to keep it close
to UT. UTC is the civilian time base in use today.
• A mean sidereal day is the time it takes the Earth to make one rotation relative to the vernal equinox.
0 hours, or the beginning of a sideral day begins at the vernal equinox direction. Due to the nutation
of the Earth’s spin a sidereal day varies. The vernal equinox itself precesses slowly westward relative
to the fixed stars, completing one revolution in about 26,000 years so that a sidereal day is some 0.008
seconds shorter than the Earth’s period of rotation relative to the fixed stars.
LECTURE 7. TIME SYSTEMS
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• Differences in apparent solar time around the Earth are related to the longitude of the observer. The
Earth is divided into 24 time zones approximately 15 degrees of longitude apart. The local mean solar
time in each zone differs from the neighboring zones by 1 hour. A time given in terms of a particular
time zone can be converted to Universal Time or UTC by simply adding or subtracting the correct
number of hours. For example, if it is 1800 Eastern Standard Time (EST) the Universal Time is 2300.
The conversion for time zones in the United States is as follows:
EST + 5hrs = UT
CST + 6hrs = UT
MST + 7hrs = UT
PST + 8hrs = UT
• Solar and sidereal day Figure 2.9-1 from [?] and not that there is one extra sideral day per year.
• Figure showing the relation between UT, Sideral Time, Prime Meridian over seasons. Remember!
When the Vernal Equinox is on the meridian, the sidereal time is 0 hours.
7.2
Calendar and Julian Day
Calendar dates in the form of year, month, and day are problematic time keeping systems for long durations.
They do not present a continious system suitable for space systems engineering. Instead the desire is to keep
these time keeping systems in sync with the orbital motion of the earth such that the equinoxes and solstices
occur near the same days each year. Various calendar systems have been in use at different times and places
around the world. The calendar system now used universally for civil purposes in the western world is the
Gregorian calendar. Its predecessor was the Julian calendar which has identical month names and number
of days in each month differing only in the rule for leap years. The Julian calendar has a leap year every
fourth year, while the Gregorian calendar has a leap year every fourth year except century years not exactly
divisible by 400. The changeover from the Julian calendar to the Gregorian calendar occurred in October of
1582, according to a scheme instituted by Pope Gregory XIII. The omission of ten days of calendar dates was
necessitated by the astronomical error built up by the Julian calendar over its many centuries of use, due to
its too-frequent leap years. The proximate problem was that that Earths seasons were drifting relative to
the expected calendar dates.
Some important relations for Julian dates:
• The time scale that is the basis for Julian dates is Universal Time, and that 0h UT corresponds to
a Julian date fraction of 0.5. The Julian Day begins at noon whereas the calendear day begins at
midnight.
• The addition and subtraction of calender dates is awkward. The standard is to use Julian Date (JD)
which is the interval of time in day and fraction of a day since 4713 B.C. January 1, Greenwich noon.
The advantage of picking a date so far in antiquity is that practically any recorded observation of a
celestial event is a positive Julian date.
• Julian date can be computed from calender dates using integer-truncation arithmetic:
JD
=
K − 32075 + 1461 ∗ [I + 4800 + (J − 14)/12]/4
+367{j − 2 − [(j − 14)/12] ∗ 12}/12
(7.1)
−3 ∗ {[I + 4900 + (J − 14)/12]/100}/4
where I is year, J is Month, and K is day of month.
• Other beginning dates are in common usage: 0 hr UT on Sept. 17 1957 (October 4, 1957, Sputnik 1,
the first artificial satellite, was launched, thus beginning the Space Age.) and 0 hr UT on Jan 1 1950.
The first system is called the Julian Day for Space (JDS) and the second is called the Modified Julian
Day (JDM). They can be calculated from the Julian date by
JDS =
JDM =
JD − 2436099.5
JD − 2400000.5
LECTURE 7. TIME SYSTEMS
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• Software typically uses some type of Julian day for internal time calculations. Microsoft Excel uses a
system starting on January 1, 1900 at 0 UT. The conversion of a Excel data to the Julian day involves
adding 2454274.5 to the excel day number. This is the or Julian day corresponding to its base date.
This formula only works for dates after 3/1/1901 and in workbooks that use the 1900 date system.
7.3
Equation of the Equinoxs
The relation between the geocentric equatorial coordinate system and the earth fixed coordinate system is a
simple rotation about the Z-axis by some angle that increases with time. The fundamental relation for some
celestial object is
(7.2)
θG = θGo + ωe (t − to )
where to is time at which θGo is known (the epoch of θGo ) and where ωe is the Earth’s rotation rate which is
(7.292115 ± 0.0000001)−5 rad/s. Due a number of effects including the nutation of the Earth’s spin axis and
the precession of the equinoxes (the reference direction for θGo ) all of the constants in the previous equation
change with time. The precise computation of θG is part of the problem refered to as the Equation of the
Equinoxs which is tabulated in The Astronomical Almanac to great precison. For historical reasons this
motion it is described and tabulated in terms of sideral time.
The conversion of sidereal time to an angle is done by reconizing that 24 sidereal hours corresponds to
360 degrees of rotation. If the hour angle is measured with respect to the true equinox, apparent sidereal
time is being measured. If the hour angle is measured with respect to the mean equinox, mean sidereal time
is being measured. When the measurements are made with respect to the meridian at Greenwich, the times
are referred to as Greenwich mean sidereal time (GMST) and Greenwich apparent sidereal time (GAST).
Given below is a simple algorithm provided by the US Naval Observatory1 for computing apparent
sidereal time to an accuracy of about 0.1 second, equivalent to about 1.5 arcseconds on the sky. The input
time required by the algorithm is represented as a Julian date (Julian dates can be used to represent Universal
Time.)
Let JD be the Julian date of the time of interest. Let JD0 be the Julian date of the previous midnight
(0h) UT (the value of JD0 will end in .5 exactly), and let H be the hours of UT elapsed since that time.
Thus we have
JD = JD0 + H/24
For both of these Julian dates, compute the number of days and fraction (+ or -) from 2000 January 1,
12h UT, Julian date 2451545.0:
D
=
JD − 2451545.0
D0
=
JD0 − 2451545.0
Then the Greenwich mean sidereal time in hours is
GMST = 6.697374558 + 0.06570982441908D0 + 1.00273790935H + 0.000026T2
(7.3)
where T = D/36525 is the number of centuries since the year 2000; thus the last term can be omitted
in most applications. It will be necessary to reduce GMST to the range 0h to 24h. Setting H = 0 in the
above formula yields the Greenwich mean sidereal time at 0h UT, which is tabulated in The Astronomical
Almanac.
The following alternative formula can be used with a loss of precision of 0.1 second per century:
GMST = 18.697374558 + 24.06570982441908D
(7.4)
where, as above, GMST must be reduced to the range 0h to 24h. The equations for GMST given above
are adapted from those given in Appendix A of USNO Circular No. 163 (1981).
1 http://aa.usno.navy.mil/faq/docs/GAST.php
LECTURE 7. TIME SYSTEMS
7.4
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Atomic Clocks and GPS
The current trend in precision position and coordinate and time reference for space systems engineering is
from the Global Positioning System or GPS. The Navstar Global Positioning System (GPS) is a satellitebased global radio-navigation system developed, deployed, and operated by the United States Department
of Defense. GPS consists of a constellation of 24 or more satellites that orbit the Earth every 12 hours in six
orbital planes (nominally 4 spacecraft per orbit) spaced 60 degrees apart and inclined 55 degrees relative to
the equator. Between 5 and 8 GPS satellites are visible from every point on the Earth at any given moment.
Each Block II/IIA satellite contains two cesium and two rubidium atomic clocks. Each Block IIR satellite
contains three Rb atomic clocks.
GPS time is given by its Composite Clock (CC). The CC or ideal clock is derived from all of the ground
and satellite operational atomic clocks. The GPS epoch is 0000 UT (midnight) on January 6, 1980. GPS time
is not adjusted and therefore is offset from UTC by an integer number of seconds, due to the insertion of leap
seconds in UTC. The offset is given in the navigation message broadcast by each satellite and commercial
receivers apply the correction automatically.
GPS offers two modes:
• The Standard Positioning Service (SPS) provides positional accuracy (95%) of 100m horizontally and
156 meters vertically, and time accuracy (95%) of 340 nanoseconds, depending upon the receiver.
• The Precise Positioning Service (PPS) provides a positional accuracy (95%) of 22 meters horizontally
and 27.7 meters vertically, and time accuracy (95%) of 200 nanoseconds.
Differential GPS (DGPS) methods are employed when higher positional precision is required. By making
multiple measurements over time, sub-meter (centimeter or even millimeter) precision is achievable using
linked receivers and sophisticated statistical analysis methods.
7.5
An Example
An example of transforming a vector from a (up, east, north) coordinates to Geocentric coordinates and the
inverse transformation.
• Consider the location of some facility on the surface of the Earth such as a satellite ground station with
location given by geodetic latitude, θlat , and longitude, θlong . At this site we would have a local (up,
east, north)
• The angle between the unit vector I (vernal equinox direction) and the Greenwich meridian is called θg
the ”Greenwich sidereal time.” If we let θlong be the geographic longitude of the site measured eastward
from Greenwich, then
θ = θg + θlong
where θ is called ”local sidereal time.”
• The angles θlat and θ completely determine the relationship between the Earth-fixed coordinates and
the Geocentric coordinates.
• If we knew θg (to ) at some particular time, to (say Oh Universal Time on 1 Jan 1996) we could determine
θ at any time t from
θ = θg (to ) + ω⊕ (t − to ) + θlong
where ω⊕ is the angular velocity of the earth.
• The Astronomical Almanac lists the value of θg at Oh UT for every day of the year.
• We now determine the rotation matrix that transforms a vector from (up, east, north) to geocentric
components. Rotation about z followed by a rotation about y.