Astronomy - synodic/sidereal periods

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In summary, the problem involves finding the distance of Mars from the sun in AU units using its synodic and sidereal periods, as well as its elongation angles on two specific dates. A careful diagram is necessary to visualize the problem. The solution involves using the law of sines and cosines to find the length of Mars' arc in terms of AU.
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Homework Statement


Mars has a synodic period of 779.9 days and a sidereal period of 686.98 days. On 2/11/1990 Mars had an elongation of 43-deg W. The elongation of Mars 687 days later on 12/30/1901 was 15-deg W. What is the distance of Mars from Sun in AU units.
(Hint: This is a multi-step problem that requires a carefully drawn diagram. You are not allowed to use Kepler's 3rd Law.)


Homework Equations


[itex]\frac{1}{Synodic_Mars} = \frac{1}{Sidereal_Earth} - \frac{1}{Sidereal_Mars}[/itex]
Law of sines
Law of cosines

The Attempt at a Solution


My drawing consists of the Sun, Mars, Earth, and the planetary orbits. From a top view, I have the situation starting with Earth at the bottom of its orbit and Mars is more left in its orbit so that the Sun-Mars line and the Mars-Earth line make a 43-degree angle. Then I draw Earth's path as it makes almost 2 revolutions but stops to the left of Earth's original position, and Mars is still in its same place it was before, os now the Sun-Mars line and the Mars-Earth line make a smaller 15-degree angle. So now I have this little arc of space between Earth's two different spots in its orbit, I know that the Sun-Earth line's value is 1 AU, I have these two angles, and I know the synodic and sidereal periods.

I just am not sure how to start calculating anything now. I've been told by a classmate that I should use the law of sines first to get as much information as I can and then use the law of cosines at the end to find the Sun-Mars distance but I am not sure how to begin. Could anyone tell me the direction I need to be going in? Thanks!
 
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  • #2
Between the two observations, both planets traverse some parts of their trajectories. You know the length of Earth's arc in terms of AU. The one of Mars can also be expressed in terms of the "Martian AU".

The two observations also form some triangles that could be used to figure out the length of Mars' arc in terms of AU.
 
  • #3
I guess this problem is hard to get help with over the internet with no drawings but I just wanted to add that I finally got it - it was just geometry. I just made sure I drew my diagram correctly then used law of sines to get all the angles and sides I needed then used the law of cosines once at the end to find the final distance.

Thanks!
 

1. What is the difference between synodic and sidereal periods in astronomy?

The synodic period is the time it takes for a celestial body to return to the same position in relation to another body, while the sidereal period is the time it takes for the body to complete one full orbit around its parent body. In simpler terms, the synodic period is the time it takes for two bodies to line up again, while the sidereal period is the time it takes for one body to make a complete trip around another body.

2. How are synodic and sidereal periods calculated?

Synodic and sidereal periods are calculated using Kepler's third law of planetary motion, which states that the square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit. This means that the longer the orbital period, the greater the distance between the two bodies.

3. What is the significance of synodic and sidereal periods in studying celestial bodies?

Synodic and sidereal periods are important in understanding the movements and relationships between celestial bodies. They help scientists predict and track the positions of objects in the sky, and can also provide insight into the physical characteristics and behaviors of these bodies.

4. Can synodic and sidereal periods change over time?

Yes, both synodic and sidereal periods can change over time due to various factors such as gravitational interactions with other bodies, changes in the orbit of the parent body, and external forces acting on the celestial body.

5. How do synodic and sidereal periods affect the perception of celestial events on Earth?

Synodic and sidereal periods can affect the perceived timing and frequency of celestial events on Earth. For example, the synodic period of the Moon and the Sun is approximately 29.5 days, which is why we experience a full moon every month. The sidereal period of the Moon is 27.3 days, which is why we see the Moon in a slightly different position in the sky each night.

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