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17. Simplified Planetary Motion
The Earth’s orbit around the Sun defines the plane of the
ecliptic. Therefore, the Sun and Earth both lie exactly on the
plane of the ecliptic, and equivalently the Sun is seen by
definition to lie exactly on the ecliptic as viewed from the Earth.
The other planets of the solar system lie approximately but not
exactly on the ecliptic: their orbits lie on planes which are at an
angle to the ecliptic plane. This angle is called their orbital
inclination i.
The Earth’s orbit is also not perfectly circular: it is an ellipse,
whose deviation from a true circle is characterised by its orbital
eccentricity e (see Dynamical Astronomy). Similarly, the other
planets have their own eccentricities.
Together, these factors make apparent planetary motion rather
complicated. However, for all the major planets of the solar
system, the values of i and e are reasonably small: the largest in
both cases is for Mercury, with i = 7.0° and e = 0.21, and for all
the other planets i does not exceed 3.5° and e does not exceed
0.1. To a reasonable approximation, we can therefore make the
assumptions that
i) all planets have i = 0, i.e. they lie exactly on the ecliptic,
ii) all planets have e = 0, i.e. they have perfectly circular
orbits (including Earth).
These simplifying assumptions make planetary motion
substantially more straightforward to model. However they will
limit the accuracy of our calculated planetary positions when
compared to the true positions of the planets in the sky.
As the planets orbit the Sun, we view them from the Earth,
which is also in orbital motion around the Sun, i.e. we see the
other planets from a moving vantage point. Therefore, planetary
position calculations will also be more tractable if we initially
adopt the heliocentric viewpoint.
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Planets closer to the Sun than Earth is (i.e. Mercury and Venus)
are known as inferior planets. Conversely, planets further
away from the Sun than Earth is (i.e. Jupiter, Saturn, Uranus,
Neptune) are known as superior planets.
When a planet appears in the same direction as the Sun when
viewed from Earth, it is said to be in conjunction. Superior
planets have only one conjunction. Inferior planets have two
conjunctions, one when they are in front of the Sun, called
inferior conjunction, and one when they are behind the Sun,
called superior conjunction. When superior planets are in a
position directly opposite to that of the Sun, they are said to be
in opposition. Planets at conjunction or opposition lie on a
straight line connecting the planet, the Earth and the Sun. When
the line joining a superior planet and the Earth makes a right
angle with the line joining the Earth and the Sun, the superior
planet is said to be in quadrature.
superior planet in opposition
superior planet in conjunction
superior planet in quadrature
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Lecture 9
inferior planet in
inferior conjunction
inferior planet in
superior conjunction
The angle between the Sun-Earth line and the Earth-planet line
is called the elongation η of the planet. (A superior planet at
quadrature therefore has η = 90°.) The angle between the Sunplanet line and the Earth-planet line is called the phase angle φ
of the planet. Working in AU, the distance from the Earth to the
Sun is 1, and the distance from the Sun to the planet is the
planet’s orbital radius a.
a
η
φ
1
Depending on the relative
positions of the planet and
the Earth in their orbits,
superior planets can have any
value of elongation η
between 0 and 180°.
Inferior planets will reach a
maximum elongation when
the Earth-planet line is
tangential to their orbit. At
this point, their phase angle
φ = 90°.
A1 Positional Astronomy
a
φ
η
1
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Lecture 9
P
φ
a
η
S
E
1
Consider the Sun-Earthplanet triangle, i.e. plane
(flat) triangle SEP for any
planet P at an arbitrary
elongation η.
We can apply the sine formula for plane triangles (equation 6.5)
to triangle SEP:
p
e
=
sin P sin E
1
a
=
sin φ sin η
sin η = a sin φ
[17.1]
Equation (17.1) is true for all planets at any point in their orbit.
In the case of an inferior planet at maximum elongation, φ = 90°.
Then the equation reduces to
( )
sin ηmax = a sin 90o
sin ηmax = a × 1
ηmax = arcsin a = sin −1 a
For Mercury,
For Venus,
[17.2]
a = 0.387 AU, so ηmax = 22.8°
a = 0.723 AU, so ηmax = 46.3°
Therefore, we would expect to observe Mercury and Venus
always to be relatively near the Sun, with Mercury never more
than 22.8° and Venus never more than 46.3° away from the Sun.
This is in reasonable agreement with the actual measured values
of ~ 28° for Mercury and ~ 48° for Venus (see Lecture 1).
Venus at western elongation is sometimes called the ‘morning
star’ and at eastern elongation the ‘evening star’.
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ωP
ωE
1 planetary sidereal
period later
Suppose the Earth orbits the Sun with a constant angular
velocity (i.e. change in angular position per unit time) of ωE, and
another planet orbits with a constant angular velocity of ωP.
The time taken for the planet to complete one full orbit relative
to the distant fixed stars (i.e. its ‘true’ orbital period) is called its
sidereal period. However, during this time, the Earth has also
been moving in its orbit: after one planetary sidereal period, the
planet still appears to be in a different position relative to the
Sun, as viewed from the Earth.
The planet’s apparent position relative to the Sun is more useful
than its position relative to the fixed stars, as its observability
depends on its angular distance from the Sun (since it can only
be observed at night). We therefore want to know how long it
takes the planet to return to its starting position relative to the
Sun, as viewed from Earth. This is called its synodic period S.
ωR
A1 Positional Astronomy
Consider a heliocentric coordinate system which rotates at
the same rate that the Earth
orbits the Sun. In this system,
the Earth is fixed in position.
The planet’s apparent angular
speed relative to these coordinates will be ωR.
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Lecture 9
The planet’s synodic period will just be the time it takes to
complete a 2π radian (360°) full revolution in this co-ordinate
system:
S=
2π
ωR
[17.3]
But ωR is just the planet’s angular speed relative to the Earth, so
ωR = ωP − ωE
(We use the modulus since ωR will be positive for inferior
planets and negative for superior planets – since planets further
from the Sun orbit more slowly – but we are only interested in
the absolute value of ωR.) So
1 ωR ωP − ωE
ω ω
=
=
= P− E
S 2π
2π
2π 2π
By analogy with equation (17.3) above, (2π / ωP) will be the
planet’s sidereal period TP, and (2π / ωE) will be the Earth’s
sidereal period TE. Therefore
1
1 1
=
−
S TP TE
[17.4]
We could repeat this analysis, but instead of choosing the Earth
as the basis for the rotating co-ordinate frame, use any other
orbiting body. Therefore, more generally, the synodic period S
of a solar system object P1 with sidereal period T1 relative to
some other object P2 with sidereal period T2 will be
1 1 1
= −
S T1 T2
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[17.5]
Lecture 9