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Physics Lesson 22.3.5 - Calculating the Time Needed for the Earth and Another Planet to Realign Again
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Welcome to our Physics lesson on Calculating the Time Needed for the Earth and Another Planet to Realign Again, this is the fifth lesson of our suite of physics lessons covering the topic of The Moon's Movement. Eclipses. Calendars, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Calculating the Time Needed for the Earth and Another Planet to Realign Again
Let's select an outer planet for this task (one from Mars to Neptune). We take as initial value of time the instant when the Sun, Earth and the planet are collinear. The Earth revolves around the Sun at angular velocity
where TE is the stellar (sidereal) period of Earth revolution.
On the other hand, the other planet revolves around the Sun at angular velocity
where Tp is the stellar (sidereal) period of planet's revolution.
Given that, in this instance, we are considering an outer planet (its distance from the Sun is longer than that of the Earth), the stellar period of planet is larger than stellar period of the Earth because the angular velocity of planet is smaller than that of the Earth (the planet rotates slower than the Earth around the Sun). This means the Earth has made more revolutions around the Sun than the given planet when they align again in the same direction. Obviously, during this time the Earth has made more than one revolution around the Sun (assuming that the planet has made at exactly one revolution around the Sun).
We denote by θ the time required for this process (i.e. the synodic period of the planet). We have
where 2π is the complete angle in radians.
Simplifying both sides by 2π, we obtain
1 = θ/TE -θ/Tp
Dividing both sides by θ, we obtain
This approach can be used to calculate the realignment time of the two inner planets (Mercury and Venus) as well. The only difference is that the Earth rotates slower this time. Hence, the above formula becomes
Example 3
Assuming that today the Sun, Earth and Mars are perfectly aligned, calculate the next time when this phenomenon will occur. Take the following values for periods of revolution around the Sun: TE = 365.24 days and TM = 686.97 days.
Solution 3
Mars is an outer planet, so we must use the first formula. We have
= 1/365.24 + 1/686.97
= 0.0027379 + 0.0014557
= 0.0041936
Thus, the new alignment of planets will occur in
= 238.46 days
You have reached the end of Physics lesson 22.3.5 Calculating the Time Needed for the Earth and Another Planet to Realign Again. There are 5 lessons in this physics tutorial covering The Moon's Movement. Eclipses. Calendars, you can access all the lessons from this tutorial below.
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