# Venus Transit - What should be a simple problem

• Mazin Nasralla
In summary, the conversation discusses the calculation of the frequency of a transit between Earth and Venus if their orbital inclinations were the same. The participants share different equations and methods for solving the problem, including using the orbital periods and relative angular velocities of the planets. They also discuss the significance of the synodic period and how it relates to the mean Venereal solar day. Ultimately, they come to a simple equation that can be used to calculate the frequency of transits between the two planets.
Mazin Nasralla
I was given a question which I have worked out, but not to my satisfaction...

"if the inclination of Earth and Venus orbit was the same, how often would the transit occur?"

I am given 4 options and I have got the right answer, but I did it by trial and error, and I am sure that there must be a better solution.

I looked up Venus's Orbital Period and it's 224.7 days. Call it 224 days.

So if we start off in a transit with the planets aligned on an axis, the two planets trace out different angular displacements over time:

Earth 2Pi/365 days and Venus 2Pi/365 days. I thought that I should be able to create a simple equation and solve, but I can't. Can anyone help?

I worked out by putting days in that the answer is 581 days (maybe small error due to rounding), but I am more interested in how I should go about solving this.
Please note that I am a pre-University student so go easy on me! Thanks

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both planets start out at an angle of 0, there will be an alignment again when the difference in angles is a multiple of 2pi.

$$\frac {2 \pi } {224} t - \frac {2 \pi } {365} t = 2 k \pi$$

divide out 2pi and solve for t

Thanks Willem.

I realized after posting how to do it.

The simple equation is

$$\frac{1}{\frac{1}{period_{Venus}}- \frac{1}{period_{Earth}}} = period_{synodic}$$

Where the synodic period is the time between conjunctions of Venus and the Sun.

This will be the average value, as due to the eccentricities of both Venus' and Earth's orbits, there will be some variation in the timing between actual transits. What is interesting in this is that this average value is an even multiple of the mean Venereal solar day (the time it takes Venus to rotate once with respect to the Sun.). This means is that Venus always presents the same side towards Earth when it is in conjunction ( or during a transit).

Janus said:
The simple equation is

$$\frac{1}{\frac{1}{period_{Venus}}- \frac{1}{period_{Earth}}} = period_{synodic}$$

Where the synodic period is the time between conjunctions of Venus and the Sun.

This will be the average value, as due to the eccentricities of both Venus' and Earth's orbits, there will be some variation in the timing between actual transits. What is interesting in this is that this average value is an even multiple of the mean Venereal solar day (the time it takes Venus to rotate once with respect to the Sun.). This means is that Venus always presents the same side towards Earth when it is in conjunction ( or during a transit).
Thanks that's even simpler! The first equation has pi as a factor on both sides and has a constant which could be awkward - especially for me, but it hints at where it came from.

How do you remember that, or derive it. Willem's equation makes sense. I can see where it comes from with reference to the 2pi. Your equation is simpler but it's not quite the equivalent. I don't think I can get to it from Willem's. The 2Pi is a factor on both sides but that's about it. Is it a stupid question to ask why it works?

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Mazin Nasralla said:
Thanks that's even simpler! The first equation has pi as a factor on both sides and has a constant which could be awkward - especially for me, but it hints at where it came from.

How do you remember that, or derive it. Willem's equation makes sense. I can see where it comes from with reference to the 2pi. Your equation is simpler but it's not quite the equivalent. I don't think I can get to it from Willem's. The 2Pi is a factor on both sides but that's about it. Is it a stupid question to ask why it works?

You can get to it along lines very much like Willem's

You are staring out with two planets in line with the Sun. The inner faster planet is is going to travel around the Sun and catch up with the outer planet. If the period of the inner planet is P1 then its angular velocity will be 2 pi/P1. If if the outer planet's period is P2 then its angular velocity is P 2 pi/P2. Thus their relative angular speed to respect to each other is $$\frac{2 \pi}{P1}- \frac{2 \pi}{P2}$$

Now from the inner planet's perspective, this is like having the outer planet stay still while it travels 2 pi radians and returns to the outer planet's position. The time it takes to do this at the above relative angular speed between the planets is
$$T=\frac{2 \pi}{ \frac{2 \pi}{P1}- \frac{2 \pi}{P2}}$$

2 pi cancels out on both top and bottom of the fraction, giving you the simple equation.

Janus said:
You can get to it along lines very much like Willem's

You are staring out with two planets in line with the Sun. The inner faster planet is is going to travel around the Sun and catch up with the outer planet. If the period of the inner planet is P1 then its angular velocity will be 2 pi/P1. If if the outer planet's period is P2 then its angular velocity is P 2 pi/P2. Thus their relative angular speed to respect to each other is $$\frac{2 \pi}{P1}- \frac{2 \pi}{P2}$$

Now from the inner planet's perspective, this is like having the outer planet stay still while it travels 2 pi radians and returns to the outer planet's position. The time it takes to do this at the above relative angular speed between the planets is
$$T=\frac{2 \pi}{ \frac{2 \pi}{P1}- \frac{2 \pi}{P2}}$$

2 pi cancels out on both top and bottom of the fraction, giving you the simple equation.
I like that explanation. I found a snippet on wiki under synodic period which confirms the formula.

https://en.wikipedia.org/wiki/Orbital_period#Synodic_period

Thanks very much to all who contributed. I am happy now, and I know a little bit more!

## 1. What is a Venus transit?

A Venus transit is a celestial event in which the planet Venus passes directly between the Earth and the Sun, appearing as a small black dot moving across the face of the Sun.

## 2. When does a Venus transit occur?

A Venus transit occurs when the orbital plane of Venus aligns with the orbital plane of Earth, which happens every 243 years in a repeating pattern. The last transit occurred in 2012 and the next one will occur in 2117.

## 3. How long does a Venus transit last?

A Venus transit can last anywhere from a few hours to over six hours, depending on the position of Venus in its orbit and the location of the observer on Earth.

## 4. Why are Venus transits important for scientists?

Venus transits provide scientists with an opportunity to study the atmosphere of Venus and the Sun in ways that are not normally possible. By observing how the light from the Sun changes as it passes through Venus' atmosphere, scientists can gather valuable data about the composition and structure of the planet's atmosphere.

## 5. How can I safely observe a Venus transit?

It is extremely important to never look directly at the Sun, even during a Venus transit. The safest way to observe it is by using a special solar filter on a telescope or by using indirect methods such as projecting the image of the Sun onto a white surface. It is also recommended to seek professional guidance and equipment for safe observation.

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