Солнечная система и ее тайны

Планеты Созвездия НЛО

MASS

The transits we discussed in the previous section only contain information about the relative sizes of the planet and star. Transits do not tell us anything about the mass of either the star or the planet. However, by using additional information from the wobble method, we can find out the mass of an extrasolar planet.

There is a principle in physics called "conservation of momentum." Momentum is defined as mass times velocity. Heavier, faster objects have more momentum. Conservation says that in any given system of objects, if there are no outside forces, there will be no change in momentum for the whole system.

This means that in a two-object system, such as a planet and star, if the star moves one way, the planet moves the other way. Because the star is much heavier, it moves more slowly, and the planet moves more quickly. We've discussed this before when we talked about the wobble method.

The equation for this is very simple: the momentum of the star is equal to the momentum of the planet. In mathematical terms:

If you prefer ratios, you can easily rearrange this to obtain a ratio:

If you know three of these numbers, you can simply calculate the fourth.

Imagine that we have a new star system that has been observed to have a planet using both the transit method and the wobble method. Doppler spectrometry from the wobble method tells us that the velocity of the central star is 40 meters/second. By knowing the star's brightness and temperature, we determine that its mass is 3x1030 kilograms.

By knowing how often the planet orbits its star and how heavy the star is, we can find out the planet's velocity. This planet is moving at a speed of 30,000 meters/second.

We can plug these values into our equations above:

Solving for the mass of the planet, we get a value of 4x1027 kilograms.

Bonus points: Because we have the planet's size via the transit method, we can also find the planet's density! This will tell us a lot about what the planet could be made of.

You will have an opportunity to practice this at the end of this section.

Footnote 1: We are looking at the case where the entire planet + star system is stationary. While this may seem like a special case, it is still ok for us to use it. The system may be moving from our viewpoint, but we can just as easily say that the system is stationary and we are the ones who are moving. Treating the system as stationary makes our math easier and does not make our solution any less correct.

Footnote 2: Determining a planet's speed from its orbital period and the mass of the central star uses Kepler's Laws. The mathematics are not very difficult, but we will not go into them in this class.

Солнечная система и ее тайны