The document discusses key concepts in geodetic astronomy including orbits, the celestial sphere, the ecliptic plane, and equinoxes and solstices. It provides historical context on how our understanding of orbits has developed from early geocentric models using epicycles to Kepler's elliptical model centered on the Sun. Kepler's laws established that planets follow elliptical rather than circular orbits, with speed dependent on distance from the Sun and a relationship between orbital properties and period.

1. Geodetic Astronomy
Orbits & Ecliptic
• History
• Orbit in physics
• Celestial Sphere
• The Ecliptic
• Ecliptic Plane
• Equinoxes and Solstices
• Synodic and Sidereal Periods
Ahmed Eldehdeh

2. History
 Historically, the apparent motions of the planets were first understood
geometrically (and without regard to gravity) in terms of epicycles, which are
the sums of numerous circular motions. Theories of this kind predicted paths
of the planets moderately well, until Johannes Kepler was able to show that
the motions of planets were in fact (at least approximately) elliptical
motions.
 In the geocentric model of the solar system, the celestial spheres model was
originally used to explain the apparent motion of the planets in the sky in
terms of perfect spheres or rings, but after the planets' motions were more
accurately measured, theoretical mechanisms such as deferent and epicycles
were added.
 Although it was capable of accurately predicting the planets' position in the
sky, more and more epicycles were required over time, and the model
became more and more unwieldy.
 The basis for the modern understanding of orbits was first formulated by
Johannes Kepler whose results are summarized in his three laws of planetary
motion.

3. History
 First, he found that the orbits of the planets in our solar system are elliptical,
not circular (or epicyclical), as had previously been believed, and that the
Sun is not located at the center of the orbits, but rather at one focus.
 Second, he found that the orbital speed of each planet is not constant, as had
previously been thought, but rather that the speed depends on the planet's
distance from the Sun.
 Third, Kepler found a universal relationship between the orbital properties of
all the planets orbiting the Sun.
 For the planets, the cubes of their distances from the Sun are proportional to
the squares of their orbital periods. Jupiter and Venus, for example, are
respectively about 5.2 and 0.723 AU distant from the Sun, their orbital
periods respectively about 11.86 and 0.615 years. The proportionality is seen
by the fact that the ratio for Jupiter, 5.23/11.862, is practically equal to that
for Venus, 0.7233/0.6152, in accord with the relationship.

4. Orbit in physics
 In physics, an orbit is the gravitationally curved path of an object around a
point in space, for example the orbit of a planet around the center of a star
system, such as the Solar System. Orbits of planets are typically elliptical
 Current understanding of the mechanics of orbital motion is based on Albert
Einstein's general theory of relativity, which accounts for gravity as due to
curvature of space-time, with orbits following geodesics.
 For ease of calculation, relativity is commonly approximated by the force-
based theory of universal gravitation based on Kepler's laws of planetary
motion.

5. Celestial Sphere
 The stars can be imagined to be points of light on a sphere which rotates
about the Earth.
 Projecting the Earth's poles and equator out onto this imaginary sphere
provides a framework for celestial measurement.
 Formal measurements of viewing direction from the Earth are usually
expressed in terms of right ascension and declination, the analogs to
longitude and latitude on the surface of the Earth

6. The Ecliptic
 The apparent path of the Sun's motion on the celestial sphere as seen from
Earth is called the ecliptic. The ecliptic plane is tilted 23.5° with respect to
the plane of the celestial equator since the Earth's spin axis is tilted 23.5°
with respect to its orbit around the sun.
 The ecliptic plane intersects the celestial equatorial plane along the line
between the equinoxes.
 The tilt of the Earth's axis with respect to the ecliptic is responsible for
Earth's seasons.

7. Ecliptic Plane
 If the sun's path is observed from the Earth's reference frame, it appears to
move around the Earth in a path which is tilted with respect to the spin axis
at 23.5°. This path is called the ecliptic. It tells us that the Earth's spin axis is
tilted with respect to the plane of the Earth's solar orbit by 23.5°.
Observations show that the other planets, with the exception of Pluto, also
orbit the sun in essentially the same plane.
 The ecliptic plane then contains most of the objects which are orbiting the
sun. This suggests that the formation process of the solar system resulted in a
disk of material out of which formed the sun and the planets. The 23.5° tilt
of the Earth's spin axis gives the seasonal variations in the amount of sunlight
received at the surface.
 Pluto 's orbit is exceptional in that its orbit makes an angle of 17° with the
Earth's orbit . This has led to a number of theories about Pluto's origin.
 Mercury is the only other planet which moves significantly away from the
ecliptic plane (7°).

8. Ecliptic Plane
 Still, we can think of the solar system as being quite flat. If we were to view
the planets’ orbits from a vantage point in the ecliptic plane about 50 A.U.
from the Sun, only Pluto’s orbit would be noticeably tilted.

9. Equinoxes and Solstices
 The points where the ecliptic crosses the equatorial plane of the celestial
sphere are called equinoxes. On those dates there are 12 hours each of
daylight and dark. The most northern excursion of the sun is called the
summer solstice and will have the longest amount of daylight. The winter
solstice opposite it is the shortest period of daylight.

10. Synodic and Sidereal Periods
 The period of a planet's orbital period around the Sun with respect to the distant
stars is called its sidereal period. The sidereal period of the Earth is about 365 1/4
days. Another type of period is useful for viewing the other planets - the period
between the times their positions both lie on the same radial line from the sun,
called the synodic period. When planets are on the same radial line from the sun,
they are said to be "in opposition". For planets closer to the sun than the Earth,
the synodic period of the Earth is longer than the sidereal period, and for outer
planets it is shorter if seen by an observer on those planets.
 The sidereal period of Mars is 1.88 years, whereas the synodic period is 2.135
years as seen from the Earth. The time of opposition of Mars is associated with its
apparent retrograde motion for an Earth observer.
 The period which brings the back to the same angular position with respect to the
Sun is called the tropical year and is 365.242 mean solar days. Formally this period
is defined as the interval of time from one vernal equinox to the next. The
sidereal period (period with respect to the distant stars) of 365.256 mean solar
days is about 20 minutes longer because of the precession of the Earth's spin axis.
That precession period of about 26,000 years brings the vernal equinox about 20
minutes earlier each year.

11. Synodic and Sidereal Periods
 The fact that the year is not exactly 365 days has led to the inclusion of the
leap year days and other adjustments to the calendar.
 The sidereal day, which brings a "fixed" star back to the same position on the
next night, is 23 hours 56 minutes and 4 seconds.
 The practical observable effect is that stars rise about 4 minutes earlier each
night, about 2 hours earlier in a month, and appear as a parade that
progresses westward across the night sky.

12. References:
 Blatt, Frank J.,Modern Physics, McGraw-Hill, (1992)
 Boynton, W. V., et al., "Distribution of Hydrogen in the Near Surface of
Mars: Evidence for Subsurface Ice Deposits", Science 297, 81, 5 July 2002
 Krauskopf, K and Beiser, A, The Physical Universe, 7th Ed, McGraw-Hill,
1993.
 Encyclopedia Britannica, 1968, vol. 2, p. 645