The Minus
Astronomy
Predicting the Mars Oppositions - I
- Sidereal period and Synodic Period -
Varun Rayamajhi
Bhaktapur, Nepal
April 5, 2022
Contents
1 Introduction
1
2 Sidereal Period
1
3 Synodic Period
2
4 Synodic Period Relations
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4.1
Synodic Period v/s Sidereal Period . . . . . . . . . . . . . . . . . . . . . . .
7
4.2
Synodic Period v/s Semi-Major Axis . . . . . . . . . . . . . . . . . . . . . .
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5 Conclusion
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References
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SIDEREAL PERIOD AND SYNODIC PERIOD
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Varun Rayamajhi
Introduction
Have you ever wondered why the time between two new moons (or full moons) is longer
than the revolution period of the Moon around the Earth? If you are or were intrigued by
that time difference, you might have encountered the terms “Sidereal Period” and “Synodic
Period”. Just in case if you haven’t, here follows a brief discussion of the terms.
2
Sidereal Period
Figure 1: Orbit of the Earth around the Sun
Let’s assume that the Earth is at position A in its orbit around the Sun at some instant as
shown in the figure 1. We know the Earth revolves in the counterclockwise direction in its
orbit, and hence it returns to position A after some time.
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Figure 2: One complete revolution of the Earth around the Sun
The sidereal period is the time it takes for the Earth to return to position A with respect to
the fixed stars. Isn’t it the same as Earth’s orbital period? For celestial objects in general
the sidereal orbital period (sidereal year) is referred to by the orbital period, determined
by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun,
relative to the fixed stars projected in the sky.[1] The same goes for other planets or any
bodies revolving around the heavier body. So, the sidereal period is the time it takes for a
body to complete one revolution around another body relative to the fixed stars.
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Synodic Period
When I was first studying about synodic period, the topic ran above my head. I hope
this won’t be the case for the readers of this article. The synodic period is the time it
takes for an astronomical body to return to the same position with respect to the other two
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bodies (typically the Sun and the Earth). Generally, the synodic period is thought of as the
time between two successive oppositions for superior planets1 , and for inferior planets2 , it
is the time between two successive inferior conjunctions (or superior conjunctions). While
conjunctions and especially, oppositions are the matters of topic for the next article, the
opposition is discussed below to help explain the synodic period.
Figure 3: Mars in opposition to the Sun
Let us consider the orbits of the Earth and Mars around the Sun. In figure 3, the Earth is
at position A and Mars is at position B. Here, Mars is in opposition to the Sun. Literally,
opposition should mean something opposite. Yes, that’s what it is. Imagine drawing an
arrow from the Earth to the Sun and another one from the Earth to Mars. Which direction
would they point? Opposite to each other, right? It implies that, for an observer on Earth,
the Sun and Mars are in opposite directions (180◦ ): when the Sun sets in the west, Mars
rises in the east. This is the moment when Mars is said to be in opposition to the Sun
relative to the Earth. In positional astronomy, two astronomical objects are said to be in
1
2
Superior planets are the planets whose semi-major axis is larger than that of the Earth.
Inferior planets are the planets whose semi-major axis is smaller than that of the Earth.
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opposition when they are on opposite sides of the celestial sphere3 , as observed from a given
body (usually Earth).[2]
Figure 4: Revolution of the Earth and Mars around the Sun
Now, let’s focus on the synodic period. The Earth and Mars are revolving around the Sun.
We know that the Earth returns to position A after one complete orbit. During this time,
Mars reaches, let’s say point P, in its orbit as shown in figure 4. Why point P? Why not
point B? Maybe it has something to do with its orbital radius and/or orbital velocity. Does
Kepler’s third law4 come into your mind? Try figuring it out yourself.
3
In its modern sense, as used in astronomy and navigation, the celestial sphere is an imaginary rotating
sphere of gigantic radius, concentric and coaxial with the Earth. All objects in the sky can be thought of as
lying upon the sphere.[3]
4
Kepler’s third law states that the square of the period of a planet around the Sun is directly proportional
to the cube of its semi-major axis.
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Figure 5: Mars in opposition to the Sun once again
CAUTION: Your eyes might get tired while looking at figures 3 and 5 repeatedly. The
figure 6 is the combination of figures 3 and 5. Consider looking at figure 6.
Figure 6: Motion of the Earth and Mars during synodic period
After some time, the Earth will be in position C and Mars will be in position D as shown in
figure 5. This is also an opposition, isn’t it? By definition, the time that it took for Mars to
reach the same positions relative to the Earth and the Sun (i.e from figure 3 to figure 5) is
called its synodic period. But, wait, the Earth and Mars are in different positions in figure
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3 and figure 5. Shouldn’t the Earth and Mars be in positions A and B respectively for the
successive opposition? It’s also an opposition, but not the successive one. Remember that
I have mentioned “same position relative to the two bodies” while defining synodic period.
In figure 3 and figure 5, for an observer on Earth, Mars appears to be in opposite direction
to the Sun, and hence, Mars is in the same position relative to the Earth and the Sun. Try
visualizing the position of Mars in both figures (recall how I told you to draw an arrow from
the Earth while describing opposition). It implies that, for the synodic period, Mars must be
in the same position relative to the Earth and the Sun (to be in the successive opposition),
not in the same position in the orbit.
Now, let’s head back to the main purpose of the article: Graphical relationship between
synodic period and sidereal period. The conditions shown in figure 3 and figure 5 are
successive oppositions. During the synodic period, the Earth travels an angular distance of
2π radians (one orbit) and a few radians θ to catch up with the superior planet. Similarly,
the superior planet travels an angular distance of θ during this time. It means that the Earth
travels 2π radians more than the superior planet during the synodic period. In other words,
the superior planet completes one orbit less than the Earth. Why? Once again, Kepler’s
third law.
However, the case for Mars is quite different. During the synodic period, the Earth travels 4π
radians (2 orbits) and a few radians θM to catch up with Mars, and Mars travels 2π radians
(1 orbit) and a few radians θM . It is because the Earth has to travel more to catch up with
Mars since the orbital velocities of the Earth and Mars are quite close: 29.8 km/s and 24.1
km/s respectively. The table below shows the orbital velocities of the superior planets and
the Earth. But, the difference between the angular distance travelled by the Earth and Mars
still remains the same, ie. 2π radians.
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Planets Orbital Velocities
Earth
29.78 km/s
Mars
24.077 km/s
Jupiter
13.07 km/s
Saturn
9.69 km/s
Uranus
6.81 km/s
Neptune
5.43 km/s
Table 1: Orbital Velocities of Superior Planets and the Earth[4]
So, in general, during the Synodic period (S) of superior planet, the Earth travels an angular
distance of 2π radians more than the superior planet. Can you guess what happens in case
of the inferior planets?
4
Synodic Period Relations
When I was studying about synodic period, its equation haunted me.
While it’s not
complicated, I had a difficult time deriving the equation first time. It’s one of the reason I
chose to write about this topic in the article.
4.1
Synodic Period v/s Sidereal Period
So far now, we know that if the Earth travels an angular distance of θE radians and the
superior planet travels θS during the synodic period (S) then θE - θS = 2π.
θE − θS = 2π
By the relation, angular velocity =
angular displacement
, we get the following:
time
θE = ωE · S
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(1)
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θE = ωE · S
Here, ωE and ωS are the angular velocities of the Earth and the superior planet respectively.
From equation (1),
ωE · S − ωS · S = 2π
(2)
We can also write angular velocities as:
ωE =
2π
TE
ωS =
2π
TS
where,
TE = sidereal (orbital) period of the Earth
TS = sidereal (orbital) period of the superior planet
Substituting these values in equation (2) and solving, we get the following relation for the
synodic period (S) of the superior planet.
1
1
1
=
−
S
TE
TS
(3)
Equation (3) is for the superior planets, but what about the inferior planets? The process is
similar, but be careful when deciding which planet (the Earth or the inferior planet) travels
more angular distance. If you work out with the inferior planet, you will get a slightly
different equation.
1
1
1
=
−
I
TI
TE
where,
I = synodic period of the inferior planet
TI = sidereal period of the inferior planet
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(4)
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If you know that the Moon is said to be in opposition to the Sun during full moon (and
in conjunction during new moon), you can now figure out the answer to the question with
which I started the article.
Initially, equation (3) and equation (4) seemed normal to me, but when I noticed that there’s
a constant in these equations, i.e TE = 1 year, I immediately graphed these equations, and
found them interesting near the Earth. If we use TE = 1 year in the above equations, we get
S=
TS
TS − 1
(5)
I=
TI
1 − TI
(6)
With the help of graphs, I could find the synodic period of the planets if I knew only one
variable - sidereal period. Here are the graphs of equation (5) and (6) with y-axis representing
synodic period (in years) and x-axis representing sidereal period (in years).
Figure 7: Synodic Period v/s Sidereal Period for the superior planets
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Figure 8: Synodic Period v/s Sidereal Period for the inferior planets
From these graphs, we can find the synodic periods of the planets if we know about the
sidereal periods. What I found interesting in the first of these graphs is that the synodic
period is smaller for the farthest planets. But the case is opposite for the inferior planets.
These curves are the functions of sidereal period, but they also appeared to be changing
when we go further (to the outer planets). Why wouldn’t they? It’s Kepler once again. This
made me realize the synodic period can be expressed as a function of distance of the planet
from the Sun.
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4.2
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Synodic Period v/s Semi-Major Axis
I applied Kepler’s third law in equation (5) and (6), and got the following equations:
a3/2
a3/2 − 1
a3/2
I=
1 − a3/2
S=
(7)
(8)
where,
a = semi-major axis of the planets in Astronomical Unit (AU)
Try doing it on your own. It’s simple.
Here are the graphs of equations (7) and (8) with y-axis representing synodic period (in
years) and x-axis representing semi-major axis (in AU).
Figure 9: Synodic Period v/s Semi-major axis for the superior planets
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Figure 10: Synodic Period v/s Semi-major axis for the inferior planets
In all of the four graphs, the curves become interesting when sidereal year approaches 1
year or semi-major axis approaches 1 AU. These are the parameters of the Earth. If we
had a hypothetical planet in our solar system whose sidereal year and semi-major axis are
infinitesimally close to those of the Earth, such planet’s synodic period would appear to be
very large (infinity). What does that mean for us? By definition, synodic period is the time
between two successive conjunctions or oppositions · · · . An infinite synodic period would
mean that the planet will never reach its successive conjunction or opposition. In fact, if
such planet existed - although it might not be possible in reality - would appear to move
along with us since our orbits would almost overlap with each other and the sidereal period
would also be nearly 1 year and hence, nearly equal orbital velocities.
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SIDEREAL PERIOD AND SYNODIC PERIOD
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Varun Rayamajhi
Conclusion
In this article, we discussed about sidereal period and synodic period (especially synodic
period). While the equations are simple, the graphs look interesting if you observe them
closely. This article serves as a build-up for the next article where we will be predicting
the future oppositions of the planet Mars and those of others, too. Remember that our
prediction won’t be 100% accurate. Can you guess why? Before I end this article, I’ll leave
you with an information: the next Mars opposition is on 8th December, 2022.
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References
[1] https://en.wikipedia.org/wiki/Orbital_period
[2] https://en.wikipedia.org/wiki/Opposition_(astronomy)#:~:text=In%
20positional%20astronomy%2C%20two%20astronomical,in%20opposition%20to%
20the%20Sun.
[3] https://www.newworldencyclopedia.org/entry/Celestial_sphere
[4] https://public.nrao.edu/ask/which-planet-orbits-our-sun-the-fastest/
[5] Figures and Graphs were generated from https://www.geogebra.org/classic?lang=
en
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