Sunday, June 3, 2012

Some Math Behind the Venus Transit

If you're still in school next week (before I get a bunch of hate mail, this includes me), I recommend throwing out your lesson plan on Tuesday for the last transit of Venus in our lifetimes (unless you happen to live for another 105 years, but even if you do, your eyesight will be so poor you won't be able to see the next one anyway). Finals on Tuesday? Finals will be around next year. The next Venus transit won't be around until 2117.

What is this "transit" you speak of?

A transit of Venus is just like an eclipse, except instead of a sun/moon/Earth interaction it's a (surprise) sun/Venus/Earth interaction. In other words, Venus will be directly between the sun and Earth and will appear as a black dot on the sun if you look at the sun using an appropriate tool. 

 

Why should I care as a human being?

Because it's super cool.

Why should I care as a scientist?

Transits of Venus (and Mercury) have played significant roles in the advancement of science in surprisingly diverse ways.  We've used transits to observe that Venus has an atmosphere, determine the distance from the Earth to the sun, and to confirm Einstein's Theory of Relativity. This week scientists will observe the transit to learn more about the search for exoplanets, or planets outside our solar system. Oh, and it's super cool.

Why should I care as a mathematician?

In my own humble opinion, this is the penultimate 101qs/WCYDWT. If you're not intrigued by how humans, for example, figured out the distance from Earth to the sun, well then...you're most likely not reading this blog in the first place.

How far is Earth from the sun?

The mathematician James Gregory is cited in a few resources I found as recognizing that the distance from Earth to the sun could be calculated by observing a transit of Venus from two points on Earth that are far away from one another. Edmond Halley and Joseph-Nicolas Delisle both came up with different ways of doing this, both of which required observing the transit from different parts of Earth. In 1761 and 1769 (Venus transits come in pairs around 8 years apart with a gap of around 105 years between pairs), scientists made epic journeys to take part in this measurement, including a trip by Captain Cook to Tahiti in 1769 ( there is a great account of these endeavors in Bill Bryson's A Short History of Nearly Everything).

Here's an outline of how Delisle's method worked. Two people with accurately synched time pieces (another challenge in the 18th and 19th century) would observe the exact time that the transit began from two places on Earth. They would not see the transit start at the same time because of a phenomenon called parallax.

At some point, the red observer will see the Venus transit begin and will mark the time.

At a later point (because the Earth is spinning and Earth and Venus are orbiting around the sun), the blue observer will see the transit begin.

 Using Kepler's Third Law and some high school geometry, you can determine the distance Venus has orbited in this short amount of time. This leads to the absolute distance from Venus to the sun. We already knew the relative distances of the planets (a mental exercise I'm currently trying to figure out without using outside resources), which is where the astronomical unit came from. Once you know the distance from Venus to the sun, you can determine the distance from Earth to the sun using--bam--ratios.



A simplified version where you ignore the rotation and revolution of the Earth and compute the time difference solely based on the movement of Venus should be accessible to any geometry student.



If you want, you can also figure out how scientists knew the distance between the two observers on Earth in the 18th century (remember, this is long before the days of global positioning satellites).


You can actually take part in a modern version of this using the following Venus transit phone app. I believe this app uses Halley's method, which compares the length of the transit in different places, but I'd need to go out and buy a smart phone to confirm this.
 

Why are transits 8 years apart, and then more than 100 years apart?


Another great math question, this one accessible to middle school students. Here's a nice chart of past Venus transits.
The Venus 243 Year Cycle

This came from an article called The Venus Transits: The Pentagonal Cycle of Venus which was helpful for me to work out some questions I still had after trying to figure this question out myself.

This video is a good place to start to understand what needs to happen for a transit to occur. It does not however, really explain why we have this weird cycle of pairs of transits 8 years apart, separated by over 100 years without a transit. I spent some time this weekend trying to figure out why this happens, to mixed results. This articleConsider the following simpler example.

Imagine two planets, Sixer and Decca. On Sixer, it takes 6 days to revolve around the sun. On Decca, it takes 10 days (so according to Kepler et al, Decca must be further away from the sun).
The red planet is Sixer and the green planet is Decca.
Because the two planets are tilted relative to one another, there are only 2 points in their orbit where a transit can occur.


If we mark the days, this means that a transit will only occur in two situtations:
  • when both planets are on day 1 of their year 
  • when Decca is on day 6 of its year and Sixer (the red planet) is on day 4 of its year

These situations are highlighted below.

With some work using the least common multiples of the year lengths (and/or some modular arithmetic), you can determine that the two planets will reset (both celebrate the new year at the same time) after 30 days, which is also known as the least common multiple. There will be a transit on days 1, 16, 31, 46, 61, etc. This means the number of days between transits will always be 15 days (or halfway through the full cycle).
 

It turns out that with this model you'll always end up with transits that are evenly spaced. Venus and Earth work in somewhat the same way, the numbers are just not as nice and the relationship is more complicated. First of all, we actually orbit the sun in an ellipse, not a circle. We also move continuously, not in the discrete steps described by this model. I'm going to continue to ignore these issues though, and I don't know how much this affects the actual transit times. Another issue is that the planets don't have to be perfectly aligned since the sun is (relatively) ginormous. To explain what I mean by this, consider two new planets (Hex and FirtyFi) with years of 6 days and 35 days respectively.  Again, a transit will occur when both planets are on day 1 of their year. In this case, though, that is the only time the planets will be perfectly aligned. Let's assume, though, that when Hex is on day 4, the planets are close enough to being aligned that a transit will occur if FirtyFi is on day 18 or 19 (again, remember that because the two planets don't orbit in the same plane these 2 spots are the only times that an alignment will lead to a transit).

In this case, the planets will have a full cycle length of 210 days (35*6) since the year lengths are relatively prime (again, this is the least common multiple). The transit cycle lengths will now be irregular, though, occurring on day 1, 88, 124, and any multiple of 210 plus these numbers. This means the time between transits would cycle between 87, 36, and 87 days.


Try some different year lengths and restrictions to see when the transits occur. If you want to get into some serious geometry, Venus orbits the Sun in 224.70069 days while the Earth takes 365.242199 days (approximately, of course). Venus's orbital axis is tilted at 3.39471 degrees and you can estimate or figure out how close to 0 degrees this angle would need to be for a transit to occur.


What other math questions do you have about the transit?


Oh, the transit will begin around 3pm PDT on Tuesday. I'm planning on using a telescope with a filter and a pair of binoculars projecting an image on a piece of foam board (do some reading if you want to try this as you need to be very careful about not looking into the binoculars and also be careful about not melting your lenses inside your binoculars). Happy viewing!

2 comments:

  1. I really like the simplified model you created to help understand why the transits follow the seemingly strange pattern of separation in time. I think you could also probably make a model of the actual transit in geogebra, no? That might be particularly useful in helping students to understand how to calculate the distance from Venus to the sun.

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