ASTR 1230 (Whittle) Lecture Notes
4. MOTIONS IN THE SKY
& COORDINATE SYSTEMS
Star trails around South Celestial Pole
in 10 hour exposure.
This lecture supplement covers the the origins of the motions
of naked eye objects in the sky and the coordinate systems used by
astronomers to locate astronomical targets. You should review this
material in conjunction with Appendices A and B in the
Manual, but you are not required to know it in detail.
A. MOTIONS OF BRIGHT OBJECTS
Cyclical motions of bright objects in sky were the main
historical stimulus for the study of astronomy. We first describe
these motions as they might have been seen by ancient astronomers but
then explain them from a modern perspective.
The table below lists celestial motions which are easily detectable by
someone on the Earth without telescopes. We call these
"apparent" motions, because they can be produced by motions of
the observer as well as by the objects themselves.
||Annual (365 d)
||(i) 1 degree/day Eastward
||Monthly (29 d)
||(i) Eastward, N/S
|(ii) Phase change
||Generally Eastward, but complex
*Motions are measured with respect to the stars, except for the diurnal motion,
which is with respect to the local horizon.
Most of these motions are so slow that if you aren't a practicing
amateur astronomer, you probably aren't aware of them. The best way
to visualize them is in a planetarium or with a good computer sky
simulation program. We will use Starry Night to illustrate
them in class.
- The motions may be complicated, but they are repeatable
over periods of months to years. It was this cyclic nature that
encouraged early astronomers to seek a deeper understanding of them.
If the easily observable sky phenomena had been more random in character,
science might never have developed, and superstition might still
prevail. The famous science fiction story
describes what might become of a society whose planet was less
fortunately situated than ours (such that darkness occurs only every
B. EXPLANATION OF MOTIONS
In the rest of this lecture, we explain the diurnal motion and the
Sun's motion from a modern scientific perspective.
It took many centuries for astronomers to arrive at the correct
interpretation described here. Ancient Greek astronomers understood
most of this after several hundred years of work, but the knowledge
was lost and only rediscovered during the Renaissance, 1300 years
The key to complete understanding of celestial motions was introduced
by the Greeks: mathematics.
The apparent motions of celestial objects can be produced by
two entirely different effects:
The Greeks built mathematical
models of the sky based on the concepts in plane and
spherical geometry which they had developed. These reduce a
bewildering array of complex phenomena to a much simpler set
of mathematical concepts.
The apparent motions we discuss in this lecture are all in the
We observe the universe from a spherical,
- Intrinsic motions of the objects themselves with respect
to one another
- The motion of the observer, or the platform on which
he/she is standing---in this case, the Earth
spinning, moving platform.
It is difficult for most people to visualize this situation. You
instinctively feel yourself to be at rest on a flat Earth, but your
instinct here is seriously misleading.
C. EFFECTS OF EARTH SHAPE AND SPIN
- Earth is a sphere.
- Night = shadow side (away from Sun). Half of Earth is always
- Daylight = sunlight scattered by Earth's atmosphere.
Daylight glare overwhelms the light of the planets and stars, so we
cannot see them when the Sun is above the horizon.
- Earth spins on its axis with respect to the stars once in 23
hours, 56 minutes (one "sidereal day"). This causes the
apparent diurnal rotation of sky.
The universe is not spinning around the Earth once per
- The spin is counterclockwise (eastward) as seen from
above N pole; ==> apparent rotation of sky is westward
- The left-hand panel of the diagram above shows Earth viewed from above its
North Pole. The Earth rotates counterclockwise in this diagram,
carrying observers with it.
- The positions where observers on the
equator are experiencing sunrise, noon, sunset, and midnight are
marked. Mean local times of day corresponding to these points are
6 AM, 12 noon, 6 PM, and 12 midnight, respectively.
- The right hand panel shows the local "horizon plane,"
which is the plane "tangent" to Earth at your location. You can see
objects above the plane but not below---at least in principle (local
features such as buildings, trees, mountains, etc. will obscure parts
of the sky).
- Note that half of the entire celestial sphere is
always above your horizon plane.
- However, the horizon plane (and
visible hemisphere) is different at each location on Earth.
- The horizon plane sweeps across the sky as Earth spins.
Astronomical objects appear to move in the opposite direction
(shown by the arrows in the figure). Objects rise over the eastern
half of the horizon plane and set toward the western half.
- By combining the concept of the horizon plane with the definition
of local time in the left-hand panel, you can visualize what
parts of the sky are observable at any given time.
D. EFFECTS OF EARTH'S MOTION IN ORBIT
- Earth is a planet moving in orbit around the Sun
- Its orbit is nearly circular (distance to Sun varies only 3%)
- Its orbit lies in a plane called the "ecliptic" plane. Seen
face-on, the orbit is technically an ellipse but deviates only
slightly from a circle. Seen edge-on, the orbit is a thin line.
- Earth orbits Sun in 365.25 days (one year) moving at ~66,000 mph.
Motion is counterclockwise as seen from above N. pole
- Stars visible at night are those "opposite" the Sun. See figures
above. The night side of Earth is that opposite the Sun. So, in May,
the constellation Scorpio will be prominent in the night sky, while in
November, it lies in the direction of the Sun and therefore is not
visible because of the atmospheric glare.
- Warning! this drawing, and most others you
will see in this course and in astronomy texts, is grossly distorted
and not to scale! In reality, the Earth's orbit is 100 times the
diameter of the Sun; the Earth is 100 times smaller than the Sun; the
stars are vastly distant from the Earth's orbit; and the stars in a
given constellation are not necessarily near one another in space.
Obviously, no one could produce or sensibly view a figure like this
drawn to actual scale.
- Also, we are viewing Earth's orbit here such that our line of
sight makes an angle of about 25o to the ecliptic plane.
From this perspective, Earth's orbit looks highly elliptical, whereas
it is nearly circular seen face-on. Without explanations, drawings
like this one help encourage the erroneous notion that the seasons are
caused by large changes in the Earth-Sun distance.
- The Earth's motion around the Sun is counterclockwise in the
drawings above. This produces an apparent eastward "motion"
of the Sun as seen from the Earth against the stellar reference
- In the drawings, the Sun in November would be seen in
projection against Libra (if you could be outside the Earth's
atmosphere), while in December, the Earth has moved such that the Sun
is seen in projection against Scorpio (which is east of
- The average "motion" of the Sun seen from Earth against the
reference frame of the stars is 1 degree per day eastward.
- "Solar" vs. "sidereal" days:
The "solar day"
is the time between noon on successive days. The daily motion of the
Earth in its orbit means that the Earth must spin a little more
than once on its axis to bring the Sun back to the point of local
noon. The extra amount is 4 minutes, on average. This accounts for
the difference between the sidereal rotation period of the Earth (23
hours, 56 min) and the average solar day (24 hours). [See illustration
of effect of Earth's motion during one sidereal day here.]
- The Sun's location on the sky as seen from Earth is confined to
the ecliptic path = the projection of the ecliptic plane on
the celestial sphere.
- The "Zodiac" is the set of constellations through which
the ecliptic path passes. (See Lecture 1.) Those are the only
constellations which have been illustrated in the drawings above. Of
course, there are 76 other constellations not shown. You should
know the names of the Zodiacal constellations.
E. EFFECTS OF TILT OF EARTH'S AXIS
North/South Motions of Solar System Objects
- The polar rotation axis of the Earth is not perpendicular to
its orbital plane. It is tilted by 23.5 degrees. See figure
below (again, exaggerated for clarity):
- The rotation axis is fixed in direction with respect to the
stars, not with respect to the Sun. This means that, as Earth
orbits the Sun, the axis continues to point in the same direction
in 3D space. See the illustration
- Thus, the tilt of the rotation axis to the Earth's orbit implies
that the Sun, as viewed from the Earth, will appear to move north and
south of the celestial equator through the year by a maximum of + or -
23.5 degrees. In the drawing above, if the North Pole is at the top,
the Sun will be seen at its most southerly position. The total
amplitude of the swing is 2 x 23.5 = 47 degrees.
- Equivalently, as seen from Earth, the path of the Sun
on the sky (ecliptic path) is inclined 23.5
degrees to the celestial equator.
- Since the Earth's Moon and the planets also lie near the ecliptic,
they will also follow paths inclined to the equator.
- The following diagram shows
the ecliptic and the equator drawn on the celestial sphere (see Lecture 1 for the definition of the
- The Sun's N/S distance from the equator as a function
of date is shown in the diagram below (click for enlagement):
- The ecliptic crosses the celestial equator at two points
- When the Sun is at an equinox, night and day are each 12 hours
long at all latitudes. The "vernal equinox" occurs around March 21,
while the "autumnal equinox" occurs around September 21.
- The Sun is at its greatest distance from the equator (23.5
degrees) at the solstices ("sun stationary," approx. June
21 and Dec 21). At these times, one hemisphere experiences its
longest day, the other its shortest.
- The Sun's N/S distance determines the hours of daylight and
the angle at which sunlight strikes Earth's surface at a given
location. It thus determines "insolation," or the amount of sunlight
incident on a unit area of the Earth's surface during 24 hours. This
differential heating is responsible for the seasons. The
"official" beginning dates of spring, summer, autumn, and winter
correspond to the Vernal Equinox, Summer Solstice, Autumnal Equinox,
and Winter Solstice, respectively.
- The change in the geographic shadow distribution caused by
the tilt is quite dramatic (even though the shadow always covers
exactly one hemisphere of the Earth). Here are two images of the way
the shadow is distributed at about 2 PM Eastern time on August 1
(left) and December 1 (right). The Earth's surface moves eastward through
the shadow. You can immediately tell from the image which latitudes are
receiving more sunlight in a 24 hour period. Click on the thumbnails
for an expanded view.
| August 1
|| December 1
- Therefore, seasons are caused by tilt, not distance to the
Sun. If the seasons were a distance effect, winter, for
instance, would occur simultaneously in both the southern and northern
hemispheres; but instead, the seasons differ by 6 months in the two
hemispheres. The Earth is actually nearest the Sun in January, one of
the coldest months in the northern hemisphere.
F. ASTRONOMICAL COORDINATE SYSTEMS
Astronomers locate objects in the sky by using a set of coordinate
systems. Each coordinate system is tied to a particular
reference frame---for instance, the local reference frame of
a given observer at a given latitude on Earth's surface at a given
time of night.
But we already know from the discussion above that the local reference
frames for different observers are different and that their orientation
in space is constantly changing as Earth rotates. So, some more universal
reference frame is needed to tie all these local frames together.
The universal frame used for astronomical observations from Earth is
based on the concept of the celestial sphere. As illustrated
below, the celestial
sphere is a geometric construct on which to display the angular
positions of astronomical objects.
It is an imaginary hollow sphere
centered on Earth. The main reference points on the sphere are the
projections of the Earth's poles and equator onto the
sphere. The primary astronomical coordinates for an object are based
on the location which that object would appear to have on the sphere
if viewed by an imaginary observer at the center of the Earth. The
sphere is imagined to spin around the Earth once a day, carrying all
celestial objects with it.
In the next section, we describe the "RA and DEC" coordinates used to
locate objects on the celestial sphere. The subsequent section
describes coordinates useful in the local reference frame of a
Professional astronomers using big telescopes usually need to know
the positions of their targets to about 1 arc-second accuracy. So,
they use the full formalism of the system described below.
In this class, we usually do not need high accuracy to locate
targets. The most important questions you need to answer in planning
You can usually answer these questions satisfactorily by using your Sky Wheels.
However, it is useful to know the background on the basic coordinate
systems described next.
- At what times of night is a target above the horizon?
- At a given time, approximately where is a target (e.g. which
quadrant: SE, SW, NE, NW)?
- What is the maximum altitude a given target can have from the horizon?
G. RIGHT ASCENSION AND DECLINATION
- The primary coordinate system on the celestial sphere is
conceptually identical to the terrestrial two-angle coordinate system
for finding places on the Earth's surface.
The terrestrial angular coordinates are
latitude, for measuring north-south distances, and
longitude, for measuring east-west distances.
- The celestial Equatorial Coordinate System likewise uses two
angles to measure positions on the celestial sphere.
Declination is analogous to latitude, while Right
Ascension is analogous to longitude.
- The two figures above show the celestial sphere as viewed from
outside. As we see in the left-hand figure, Declination
(abbreviated DEC or Greek "delta")
is measured (in degrees) north or south of the celestial equator. The
North and South celestial poles have DECs of +90 and -90 degrees,
respectively. The equator has DEC 0.
- Another visualization of the RA and DEC system is shown here.
- Right Ascension (RA or Greek "alpha") is measured as shown
in the right-hand figure above. Lines of constant RA are great
circles (also called "hour circles") drawn through the celestial
poles. The zero point of RA is the vernal equinox (the point where
the Sun crosses the celestial equator moving north), and RA
increases eastward. (The zero point is arbitary, just like
the zero point of the longitude system was arbitrarily chosen to be
- Rather than measuring in degrees, RA is measured in time
units of hours, minutes, and seconds up to one sidereal
day. Therefore, RA values run from 0 hours to
For a star on the equator, one hour of RA corresponds to 15 degrees
of arc. But because of the convergence of the lines of
constant RA toward the pole (see figure above), at other declinations
one hour of RA corresponds to fewer than 15 degrees of arc.
- Beware confusion between "minutes" and
"seconds" of time and those of arc!
The RA and DEC coordinates of the stars change only very slowly
with time and can be considered fixed for the purposes of this
course throughout the year.
The slow changes are the result of two
effects: the change (or "precession") in the direction of the Earth's
polar axis and small motions (called "proper motions") of the stars
with respect to one another. Precession is discussed
- The RA and DEC of the Sun and Moon change continuously as they
move from day to day. The Sun is always on the ecliptic, so its DEC
is always between -23.5o and +23.5o. At the
vernal equinox (near March 21), the RA of the Sun is 0 hours and at
the autumnal equinox (near Sept. 21), it is 12 hours.
Local Reference Frame for an Observer on
Earth's Surface. (Green shows the horizon plane.)
H. THE LOCAL REFERENCE FRAME
The local reference frame of any Earth-bound observer is determined by
that person's instantaneous horizon plane. The diagram above shows the local
reference frame and its main reference points.
The zenith is the point on the sky directly overhead.
The meridian in the great circle drawn on the celestial sphere
through the pole(s) and the zenith. The points where the meridian crosses
the edge of the horizon define the directions on the horizon plane to
North and South.
The points where a great circle running through
the zenith at right angles to the meridian cross the horizon plane define
the directions to East and West.
Two angles also suffice to locate any object in the local reference
frame of an observer. Engineers would use an azimuth
and altitude system, but conversion to these angles from
RA and DEC requires the use of spherical trigonometry. With telescopes
which have equatorial mounts, astronomers can use a simpler
system based on DEC and "Hour Angle."
Visibility of Astronomical Objects: Declination
It is important to know how to determine when astronomical
objects are well placed for observation from your particular location
on Earth at a given date and time. The "DEC-HA"
method is the quickest way to do this:
- The figure below shows how the Declination of an astronomical
object controls its visibility with respect to your horizon. The
shaded plane is the horizon. Star tracks are shown on the celestial
sphere. The orientation of the sphere for your location is the
same at all times of the year (though the Sun moves to
different locations along it and thus changes the part of it which is
visible at night).
- As the Earth rotates on its axis during the course of a night, the
stars appear to rise in the east, move along the heavy lines of
constant declination, and set in the west. A star is farthest
from the horizon when it crosses your meridian.
At this time it is
said to transit.
- The figure above helps you visualize how long in each 24 hour
period stars at different DECs are above your horizon. Stars at some
DECs (like +60 degrees here) are always above your horizon, 24
hours a day. Such stars are called circumpolar stars; they
never set. Most of the stars shown in the startrail image at
the top of this webpage are circumpolar stars.
- By contrast, stars at -30 degrees DEC are above the
horizon only for short periods. There are some stars farther south
which are never above your horizon. You would have
to change your latitude to see them.
Visibility of Astronomical Objects: Hour Angle
- Knowing DEC alone is not sufficient. The celestial sphere
rotates continuously about the axis through its poles and therefore
continuously changes the location of an astronomical object in your
local coordinate system. An object's DEC may always be the same, but
its east-west position changes during the night.
- To locate an
astronomical object in the sky, you therefore need to know one additional
quantity besides DEC: the distance of its hour circle from your local
- The Hour Angle or HA is defined to be the angle
measured along the equator between the hour circle of a star and your
meridian. In the diagram above, the hour angle is denoted "HA." It
is quoted in time units (hours, minutes, and seconds).
Objects to the west of the meridian have
positive HA, while objects to the east have
By convention, HA's are always in the range -12 hours to +12
A star with HA = 2 crossed the meridian (i.e.
reached transit) two hours ago, while a star with HA = -1 will be on the
meridian (will transit) in one hour.
- You must know both HA and DEC to determine visibility (refer to
the diagram at the start of this section).
- An object on the celestial equator (with DEC = 0) with HA =
-6 hours (+6 hours) is just rising (setting) on the horizon. It will
be above the horizon for 12 hours a night. It will rise due east
and set due west.
- From Charlottesville, objects south of the equator will be above
the horizon for fewer than 12 hours. They will rise south of due east
and set south of due west. They will be below the horizon even if their
HA is 6 hours east or west.
- Objects north of the equator will be above the horizon for more
than 12 hours. They will rise north of due east and set north
of due west. They can be above the horizon even if their HA's are
larger than 6 hours east or west.
- Any object in the circumpolar
region is above the horizon all the time (for any HA), but
cannot be seen in the daytime because of scattered light from the
- How do we compute the Hour Angle of an astronomical object?
To determine HA we must know the current position of the zero point of
RA, the vernal equinox. This is given by the sidereal time
or ST. ST is defined to be the hour angle of the vernal
equinox, and it is the fundamental measure of astronomical time of
day. Sidereal time runs from 0 to
- By this definition, ST is also equal to the RA
of an object now on the meridian.
- At a given ST, the HA of a star with a given RA is:
- HA = ST - RA
- Examples, if the sidereal time is now 8 hours:
- An object with an RA of 8 hours has HA = 8 - 8 = 0
is now crossing the meridian (or "transiting").
- An object with an RA of 10 hours has HA = 8 - 10 = -2 hours.
It is two hours east of the meridian and will transit in two hours.
- An objects with an RA of 5 hours has HA = 8 - 5 = +3 hours.
It crossed the meridian 3 hours ago and is now west of the meridian.
- If the numerical value of HA (apart from the sign) in this
expression is larger than 12, then add or subtract 24 hours to make
it less than 12. Examples:
- If ST is 1 hour but RA is 23 hours, then the formula gives
-22 hours. In this case, add 24 hours to obtain HA = +2 hours (i.e.
the star is 2 hours west of the meridian).
- If ST is 23 hours but RA is 2 hours, then the formula gives + 21
hours. In this case, subtract 24 hours to obtain HA = -3 hours (i.e.
the star is 3 hours east of the meridian).
- Sidereal time runs faster than the mean solar time that
our watches use by 3 minutes and 56 seconds per day, which is the
result of the Earth's orbital motion around the sun. This means that
the sidereal time at a given standard time changes continuously by
2 hours per month (or 24 hours in a year). If the ST at 9 PM
EST tonight is 6 hours, one month from now it will be 8 hours.
- It is useful to memorize the ST-local time conversion for a couple
- The ST at midnight is always the RA of the Sun plus 12 hours.
- On March 21 (the vernal equinox), at 9 PM EST the ST is
also 9 hours.
- On September 21 (the autumnal equinox), at 9 PM EST the
ST is 21 hours.
- A table listing the
ST at 9 PM EST throughout the year is given in Appendix B of the ASTR 1230
- Using the HA/DEC system and an equatorially-mounted telescope,
you can quickly locate objects in the sky if your know their RA, DEC,
and the ST. One practical problem: HA changes continuously, so when
you predict HA, you need to allow time to set the telescope! [Many modern
small telescopes use microprocessors to display coordinates in RA rather than
Altitude & Meridian Slice
- The HA/DEC system is part of the celestial Equatorial Coordinate
System defined with respect to the poles and equator. However, it is
often useful also to know how to locate an object with respect to your
- Altitude is defined to be the angle between a star
and your horizon plane. In general, you need to use trigonometry to
find the altitude of an object with a given HA and DEC.
- However, when a star transits (i.e. crosses your
meridian), there is a simple relation between its DEC, your
latitude, and its altitude. This is illustrated in the figure above,
which shows a geometric "slice" through your local meridian.
- Based on the diagram, the altitude of a transiting star above the southern
horizon (SALT in the diagram above) is given by the following
expression, where LAT is your latitude.
- LAT = 38o at Charlottesville, so here SALT = DEC +
- Objects with DEC = LAT have altitudes of
90o, i.e. they cross through your zenith. The celestial
equator (DEC = 0) has SALT = (90o - LAT).
- SALT is a useful concept for determining how high in the sky a
planet or the Moon, for instance, can be on a given night.
- As an example: the Moon orbits in a plane which is tilted
5o from the plane of the ecliptic. Since the maximum
DEC of the Sun (or the ecliptic) is +23.5o, the maximum
DEC of the Moon is +28.5o. The maximum altitude
of the Moon above the southern horizon is then:
SALT = 28.5o + (90o - 38o) =
Related Web links
September 2008 by rwo
Star trail image copyright © David Malin/Anglo Australian
Observatory. Zodiac and axis tilt drawings copyright © by Nick Strobel. Equinox, RA, &
Dec, drawings copyright © 1974,5 by Edmund Scientific Corp. Text
copyright © 1998-2008 Robert W. O'Connell. All rights
reserved. These notes are intended for the private, noncommercial use
of students enrolled in Astronomy 1230 at the University of Virginia.