Ok. So how big would a planet need to be to produce the infrared excess that we're actually seeing in Vega?
Now our working hypothesis here, so we have Vega and it's putting out radiation
and some of that radiation hits a planet
and the planet therefore becomes warm and radiates infrared radiation from its surface.
What we know is its infrared radiation here is 2.5 by 10 to the minus five times the optical radiation from here.
We also know that the infrared radiation has a spectrum that peaks at about sixty micrometres.
That's going to be crucial. Ok. How are we going to solve this?
So our first step is to work out the distance here, D, how far out the planet is.
And we can work that out by looking at the spectrum. The fact that it peaks at sixty microns is telling us something
about the temperature of the planet. You may have heard of Wien's Displacement Law
which says that the peak wavelength times the temperature equals a constant b
where b is two point nine by 10 to the minus three metres Kelvin. So in this case we know that lambda peak is sixty micrometres
sixty times 10 to the minus six metres, therefore T is b over lambda, so T equals about fifty Kelvin.
So to produce the spectrum peaking at sixty microns, we need quite a cold planet.
Now that presumably indicates that it's quite a long way away from the star.
In the first course we were talking about life in space, we worked out how hot a planet should be at different
distances, and we've repeated that since. And we find that the temperature of a planet
is equal to the fourth root of the luminosity divided by sixteen Pi Sigma D squared
where Sigma is (????) when constant. This is not perfectly accurate, it's assuming that everything
is a black body, but it will do as a first approximation. Now the luminosity of Vega is fourty times
the luminosity of the sun, and the luminosity of the sun if you remember
is three point eight by 10 to the twenty six Watts. So rearranging this, we get that D
equals root of luminosity, which is one point five by 10 to the twenty eight Watts, all
over sixteen Pi Sigma T to the fourth. Which comes out as about two point nine by 10 to
the thirteen metres, which is about two hundred astronomical units. So that's our first clue.
For a planet to be this cool it has to be a long way out. Given that it's that far out,
we can now work out how big it would need to be. So our idea is that we have Vega putting
out its luminosity, and the fraction that actually hits the planet is going to be given
by the cross-section there on the planet, divided by the surface area of a sphere of
the same radius D. So the cross-sectional area is going to be Pi r squared, where r
is the radius of the planet. We have to divide it by the area of the entire sphere, which
is four Pi D squared. And that's got to be the fraction of power that is intercepted
by the planet, and assuming that the amount radiated is the amount in so it's a steady
state, that has also got to be the ratio of infrared to optical emission. So that's going
to be our two point five by 10 to the minus five. So therefore we find that radius of
the planet equals the square root of or the Pi's cancel, take the four D squared up
here, that's the square root of everything, so it's going to be two point five by 10 to
the minus five, times four D squared which comes out as two point nine by 10 to the eleven
metres. Which is enormous, that's more than an astronomical unit. This means the planet
would have to go all the way from the Earth to the sun if it were in our own solar system.
Absolutely staggeringly big planet. The radius of Jupiter by comparison is a pathetic seven
by 10 to the seven metres. So we're talking about something that's more than a thousand
times bigger than Jupiter, way more. As it's a thousand times bigger in radius, that means
that's a thousand squared times an area, so we're talking over a million Jupiter-sized
things orbiting around at a sort of distance to give us that sort of infrared excess. So
not looking plausible. We'd need either a ridiculously stupendously large planet bigger
than any star let alone any planet we know, or millions of Jupiters
either of which seems pretty unlikely.