Radial Velocity (Super-Fast) Lab(!)

In this post I'll be doing the Radial Velocity (Super-Fast) Lab handed out in class earlier this week. We're supposed to determine the period, mass, and semi-major axis of an exoplanet, given its host star's radial velocity curve. Here are the graphs (assuming a star's mass that is 1.5 times that of the sun):

The period is easy to determine by inspection. For the first graph, the period seems to be about 1 year. We can use Kepler's Third Law to determine the semi-major axis:

So:

To determine the mass of the planet, we can use the following equation:

where K, the amplitude of radial velocity, is expressed in fractional units using a constant (29.8 m/s), the mass of the sun, a period of a year, and the mass of Jupiter. For the first graph, we have an amplitude of about 100m/s. So, our mass is:

, where M_J is the mass of Jupiter.

What about the second graph? We can see that the radial velocity amplitude K is about 50m/s, and the period is roughly 3.3 years. The semi-major axis, therefore, is:

The mass of the planet is:

Exoplanets and Wobbly Stars

Exoplanets are planets found outside our solar system, and remain the focus of a rapidly growing field of astronomy. Detecting them, however, can be difficult, since they don't radiate light like stars, and are much smaller. In this post I'll be introducing the physics of one of the methods we use to detect exoplanets. The method utilizes the fact that exoplanets exert a gravitational force on their host star, resulting in a "wobbling" of the star as it orbits ever so slightly around the two bodies' center of mass.

Let's start from the basics. Consider a star and a planet, orbiting each other around their mutual center of mass:

The equation for the center of mass is:

In this case, we have set the center of mass to be at x=0. Substituting in our variables, we have:

Simplifying, we have the following relationship:

Thus, the distance from the star/planet to the center of mass is inversely proportional to its mass.

What can we do with this? We can use Kepler's Third Law to find out how much the Sun wobbles. Kepler's Third Law:

where a is the mean semimajor axis, the sum of a_p and a_*, and M is the sum of the mass of the star and the mass of the planet. If we assume the mass of the star to be much greater than the mass of the planet, we can use Kepler's Third law for the planet's orbit and solve for either a or T (since the mass of the star is so large, a is effectively equal to the planet's orbital radius). For example, if we consider the effect of Jupiter on

the sun:

Solving for a_* (using the fact that the mass of Jupiter is about 1000th of the Sun's mass), we have:

Thus, the displacement of the Sun due to Jupiter's gravitational field is roughly a solar radius. That's an incredibly small distance to measure! That's also why we measure velocity, instead of displacement, when searching for exoplanets.

Ringworlds

After Professor Johnson's very interesting plug for Professor Jason Wright's posts on Dyson spheres, I was reminded by a science fiction novel I started (and unfortunately, never got around to finishing) a long time ago: Ringworld, by Larry Niven.

When Freeman Dyson published his idea of a Dyson sphere many years ago, he postulated that any sufficiently advanced civilization would have to harness the energy of its host star to keep on growing. The reasons for constructing a Dyson sphere are simple: they provide immense amounts of energy, and the sphere itself provides a huge amount of room to live on. Imagine a surface 1 AU in radius, all available for habitation!

Artists' rendition of a Dyson sphere. Note that, unless there's another star in the system, the sphere probably won't appear lit (since it's enveloping the host star).

The problem with Dyson spheres is that it's very, very costly to construct a sphere with a radius of 1 AU! Think of all the raw material you would have to collect.

In Ringworld, author Larry Niven poses the question of a "partial" Dyson sphere - that is, taking only an equatorial slice of a Dyson sphere, a ring, and setting it in gravitational equilibrium around the star. The amount of living space would still be much greater than that of the Earth, due to the huge radius of the Ringworld, and the ring could be set to rotate to simulate gravity. Sufficiently efficient solar collectors could be placed throughout the Ringworld to power the entire system. An atmosphere could even be created, if the Ringworld could spin fast enough to prevent the atmosphere from escaping!

Artist's rendition of a Ringworld.

The novel itself is actually pretty good, and hopefully I'll get around to finishing it one day. Ringworld was the first novel to pose the idea of humanity (or any species), living on a ring rotating around the Sun. Could humanity one day build its own Ringworld to replace Earth, and live on free solar energy?

It turns out that this isn't the case, unfortunately. Although Dyson spheres, i.e. spherical shells, are gravitationally stable, the two-dimensional ring structure of a Ringworld makes it an inherently unstable equilibrium. Back to the drawing board, then.