University of Michigan - Department of Astronomy

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Extrasolar Planets

Prelab

List the factors you think would affect the temperature on a planet's surface. Consider both how the planet can heat up and cool down. Include the effect you think each would have on the temperature (increase, decrease...).

To do this lab you will need to look at the information on these three graphs:

  1. 70 Virginis
  2. 47 Ursa Major
  3. 51 Pegasi

Objectives

Introduction

When humans look at other planets, we usually ask "could there be life there?". This life could take many forms, including some very different from our own, but because we only have information about Earth-life (carbon-based organisms) we may as well start by looking for life like us. This means we can test newly discovered planets to see if they meet certain requirements for life, the most important requirement being the presence of liquid water.

We will assume that if the temperature of the planet is right, there will be liquid water. Hydrogen and oxygen are common elements throughout space and so when planets form they usually contain water. If they don't, comets soon deliver water to the planet surface in collisions. The necessary temperature, between 0 C and 100 C, can be achieved through solar heating of the planet.

When a planet absorbs sunlight, it gets warmer. When it in turn emits blackbody radiation (mostly in the infrared -- most people think of this as 'giving off heat') it gets cooler. The planet will quickly reach an equilibrium temperature, where it gives off as much energy as it absorbs. For example, the Earth has come to an average equilibrium temperature of about 0 C (equal to 273 K).

Energy In

The power (i.e., energy per unit time) the planet absorbs is related to the amount of sunlight at the planet's location, the planet's cross-sectional area and the fraction of light that is absorbed (not reflected by the top of the atmosphere).

The amount of sunlight is proportional to the star's luminosity, L. A brighter star would put out more light, so more sunlight will reach the planet. The amount of sunlight at the planet's distance, d, from its sun is inversely proportional to d2. This is the inverse square law for light (i.e., the brightness of light decreases with d2).

This leaves:

Energy In
Eqn. 1

Energy Out

The energy given off by a planet can be determined by using the blackbody radiation concepts introduced in an earlier lab (Spectroscopy Lab). A solid, opaque body like a planet emits continuum radiation, because it is warm. The amount of energy given off per unit time can be described by by the Stephan-Boltzman Law. This says that the total energy given off from the planet's surface is proportional to the surface area (note: "surface area" = 4 "cross-section area") of the planet times the temperature to the fourth power.

This is just the energy radiated from the surface of the planet -- the dirt. Some fraction of this radiation will be trapped by the clouds and reflected back to the planet while the rest is transmitted out into space. This is also known as the greenhouse effect.

So now we also have:

Energy Out
Eqn. 2

Bear in mind that the stellar radiation absorbed by the planet and the continuum radiation emitted by the planet are peaked at different wavelengths. The star's temperature is in the thousands of Kelvin, which means that according to Wein's Law its wavelength peak emission is somewhere in the visible. The planet's temperature is generally much cooler and so its emission peaks in the infrared. This means the radiation going out "looks" different from the radiation coming in, but the total amount of energy in and out must be balanced. If the power in does not equal the power out, the planet will either warm up or cool down until balance is achieved.

light in, light out

Equating power in and power out, leads us to the following relationship for the temperature of the planet:

"energy in" = "energy out"

Balanced version 1
Eqn. 3a

Now, the planet's area cancels out on both sides, leaving:

Balanced version 2
Eqn. 3b

Note that the two sides of this equation are still not equal -- this is only a proportionality because there are constant factors that have not been included in the calculations.

Now we have a relationship between temperature, the luminosity of the star, the planet's distance from the star, and some characteristics of the planet itself (the %'s above). In order to make life a little easier, we'll assume that the % of light absorbed and the % of light transmitted on these extrasolar planets is the same as it is on the Earth. This is not exactly true, but it should be close enough for our purposes.

The temperature equation above is true for the Earth as well as for extrasolar planets. So, we can use the Earth as a comparison case. Then the constants will cancel out of the equation and we get a real equation for the temperature of the extrasolar planets

temperature eqn
Eqn. 4

where the planetary characteristics (the %'s) have been cancelled out because we assumed they were the same for the extrasolar planets as for the Earth.

The luminosity, L, can be found in a number of ways, but we will simply use the "spectral type" of the star to predict the luminosity. The spectral type just describes the star. It is kind of like categorizing lightbulbs by their wattage -- you know a 'G2' star has a certain luminosity, mass, and temperature just like you know a 60 watt light bulb will give out a certain amount of light. We will use information that astronomers have found over many years, relating the spectral type to luminsoity and mass.

The distance, d, between the star and the planet is a more difficult quantity to find. It comes out of the data that is part of finding the planet in the first place. This in itself is not an easy task. The method presently used to find involves looking for changes in the position of the star due to the orbiting motion of the massive planet. If you did the Moons of Jupiter lab, you'll remember that a planet and star actually orbit around the center of mass of the system. In most cases, the star is so much heavier than the planet that the center of mass is practically at the center of the star and the star's motion around the center of mass is very small. If the planet is heavy enough though, the star's motion can be detected. This motion, if it is along the line of sight to the star, can be detected using "Doppler Shift".

Doppler shifts are observed revealing the presence of a planet

Doppler shift is something that most of us have experienced, when a car (especially one with a siren) passes by on the highway. The sound increases in pitch (frequency) as the car comes towards us, and decreases in pitch (frequency) as the car goes away. The same thing happens with light -- as an object emitting light moves toward us, we observe the frequency to increase and as it moves away the frequency decreases. When a star is moving towards or away from us, the absorption lines in the spectra get shifted in frequency, allowing the velocity of the star along the line of sight to be calculated.

Once the velocity of the star during different parts of its orbit is measured, the orbital period of the planet can be calculated. In one period, the star should move away from us and then back towards us once. Then we assume a value for the mass of the star, which can be done in the same way we found the luminosity -- using the relationship between mass and spectral type of stars. With these two values, we can use Kepler's Third Law,

Kepler's Third Law
Eqn. 5

to find d, the average distance between the star and the planet (same as the semi-major axis). The units are listed in the above equation just to make sure you don't forget the special units Kepler's Third Law must use.


Directions

We need the luminosity of stars for the other planets, as well as the distances between the star and the planets. The spectral types of each star are listed in Table 1. Use the information given in Table 2, to convert a main-sequence spectral type to a luminosity and mass (note the units); record these values in Table 1. The observed values of velocity (derived from Doppler shift) versus time for three stars which are deduced to have planets are linked at the beginning of the lab. Find the periods of the planets from these and record this in Table 1. Use Eqn. 5 (Kepler's Third Law) to find the distance to the planet and record this in Table 1. And then, use Eqn. 4 (Stephan-Boltzman Law) to find the temperature of the planet; record this in Table 1.

If you look at the graphs of velocity versus time for the planets, they also have a quantity called 'M sin(i)' which is the mass of the orbiting planet (with the sin(i) factor to account for the fact it is impossible to tell how much the solar system is tipped from our point of view). The mass of the planet is fairly simple (in theory!) to calculate. Now you can answer the Questions on the worksheets.


Last update 9/30/05