Frontiers and Controversies in Astrophysics: Lecture 7 Transcript
February 6, 2007
<< back
Professor Charles Bailyn:
So, we've been talking about transits and I want to say a couple things more just
to round out that discussion. You'll recall how this works. Transits--you get light blocked--light from the star
blocked by the planet. And in order for this to occur, the orbit has to be exactly edge-on; otherwise, the planet
doesn't actually get in the way between you and the star. So, must be edge-on. And so, what you see,
basically, although you're too far away to see it in this kind of detail, looks something like this. Here's a star,
it's got this big, bright, shining disc, and then crossing that star is a darker disc from the planet. So, it looks
like you have a little spot there. And it's clear, just from the geometry of this situation, how much light is
obscured. It's basically the ratio of the cross-section of the planet to that of the star. So, the depth of the
transit, and by that I mean, you know, the fraction of the light that is obscured, is equal to--that will be, the
cross-section of the planet divided by the cross-section of the star. And the cross-section is just the projected
area. And so, what that means in practice is, here's the radius of the star, and there's an equivalent radius of
the planet. And it's just the ratio of the projected areas.
Now, the area of a circle, projected area of a sphere is a circle. The area of a circle, you probably remember
from geometry, is
Ï€r
2
. And som what this is going to be is
Ï€r
of the planet squared, divided by
Ï€r
of the
star squared and the
Ï€
s cancel. So, it's the ratio of the squares of the radii.
Okay, so to take an example--supposing you were an astronomer in some distant place and you're looking at
the--at our own Solar System. And you're fortunate enough to be in a place where you can see our Solar
System edge-on. What would you see when the Earth transits the Sun?
So the Earth's--so here's an example: Earth transiting the Sun. The radius of the Earth, it turns out, is
something like 7 x 10
6
meters. The radius of the Sun is about 100 times that, 7 x 10
8
meters. And so, the
depth of such a transit is the square of the ratios of these radii.
(7 x 10
6
) / (7 x 10
8
)
2
That's (10
-2
)
2
= 10
-4
, or 0.01%.
So, that's a lot less than what you found in section yesterday for a Jupiter-like planet. And the key is in this
r
2
factor, until the Earth is about 1/10--has about 1/10 the radius, is about 1/10 the size of Jupiter. But that tenth
gets squared, so that's 100. And so, instead of getting a 1% dip in the light, or approximately 1%, you get a
1/100 of 1%.
1/100 x 1% = 10
-4
.
So, as the planet gets smaller you--it becomes harder and harder to see these transits. And, if you think back
on the pictures of these light curves, the graphs that were made, in the observations from the ground, you
really needed to have about a 1% dip in order to be able to see it. From space you can do a lot better. Things
are much more stable, and you might well be able to see something like this, but only from space and not
necessarily from the ground.