non-bio homework for big science 5/11/12

7/31/2019 Non-bio homework for Big Science 5/11/12

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Big Science Non-Bio Homework

5/11/12

Johannes Kepler

1571-1630

Kepler's Third Law of planetary motion, which is the origin of Isaac Newton's theory ofgravity, states that the cube of the semi-major axis of the orbit of a planet is directly

proportional to the square of its sidereal period:

Cube: x times x times x is x cubed, also written as x3 .Semi-major axis: The mean width of an ellipse.Directly proportional: Probably best just to look at this as meaning equals for the moment,but strictly speaking, increasing or decreasing together, and with a constant ratio; - opposed to

inversely proportional.

Square: x times x is x squared, also written as x

2

.Sidereal period: The time it takes a planet to go round the Sun as seen from the fixed stars(which aren't!), in other words its year.

An example:

The mean distance of Earth from the Sun (actually from the barycentre of the Sun and Earth this will become important later) is 1 AU (Astronomical Unit), which is about 150 million

kilometres.

The mean distance of Saturn from the Sun is about 10 AU. 103=10x10x10=1000.302=30x30=900. Saturn takes about thirty years to go round the Sun. Hence the square of its

sidereal period compared to ours is directly proportional to the cube of the semi-major axis of

its orbit.


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The largest known planet in this Solar System is Jupiter (the largest unknown one beingTyche). Jupiter is 5.204 AU from the Sun on average and takes 11.86 of your Earth years toorbit it. Jupiter is so massive that it can fairly easily pull small objects out of their orbits. For

instance, if a rock took exactly half as long as Jupiter to orbit the Sun, it would be pulledslightly towards Jupiter into a new orbit every two of its orbits. Therefore, in the early Solar

System, when space around the Sun was strewn with rocks, chunks of ice and gas, Jupitercleared gaps in these where their years were close to a fraction of its own year, causing clumps

just outside those orbits where collisions were more likely to take place and planets weremore likely to form there as a result. Earth has a year just over a twelfth of that of Jupiter, and

that's probably why.

Alpha Centauri

Alpha Centauri is the nearest star system to our own. It consists of three stars: Proxima, Aand B. Proxima is the closest of all, currently about 268 000 AU from Earth, and is a red dwarf

flare star about 1.5 times the diameter of Jupiter but much more massive. It will last manytrillions of years compared to our relatively short-lived Sun. It's cooler and redder than the

Sun and may or may not have planets. It's 11 000 AU from the other two stars - as is usualwith a triple star system, the third star is more widely separated than the other two, although

the scale can be much smaller.

Alpha Centauri A and B are both like the Sun, yellow dwarfs. The name dwarf in this case ismisleading because in fact all three of these stars are more massive than about 90% of other

stars but there is no extra class between dwarf and giant stars. They are fairly similar in mass.A is 1.1 times the mass of the Sun. 52% brighter and of G2V spectral type, in other words

almost identical to our own Sun. B is cooler and smaller: 0.91 times the mass of the Sun, 50%as bright and of spectral type K1V. It's golden yellow compared to the Sun's and A's pale

yellow colour.


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Since the stars are similar masses, their barycentre is about halfway between them and theyorbit each other rather than one orbiting the other (Proxima orbits the whole system very

slowly, taking about half a million years). They take eighty years to do this and are separatedby 11 AU at their closest and 35 AU at their most distant.

In a way, B can be considered to be a gigantic planet of A's, and to a lesser extent (because it's

bigger) vice versa. A planet orbiting in a binary star system can be stable provided it is lessthan 28% of the minimum distance between the two stars or more than 3.5 times the

maximum distance.

The length of the year of a planet in the Centauri system is easy to calculate. If Earth wereorbiting where it is now relative to A, its year would be 1.1 times shorter, or 332 days, because

A is 1.1 times as massive as the Sun, and with B its year would be 400 days long because B is0.91 times as massive as the Sun.

The known planet orbiting B is at 0.04 AU from it on average and takes 3.25 days to orbit it. It

is at least 15% more massive than Earth and has a probable surface temperature of about

1500 K or 1200 degrees Celsius.

It would seem reasonable to conclude that the planets of A and B are likely to have formed just

outside where each of the companion stars has cleared a gap in the system in the same way asJupiter has cleared a gap in ours.

Therefore, assuming that this has happened, the questions are:

Where are the planets of the Centauri system most likely and least likely to be?

And:

How long are their years?

non-bio homework for big science 5/11/12

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