Download Exoplanets

The surface temperature of
exoplanets*
Physics 217 Lab
Semester 1 2015
Surname of Student:
Student No:
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
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...).
Objectives
•
•
Answer the question: Does liquid water exist on three known exoplanets?
To do this, we need to know:
• brightness of their parent stars,
• distance of exoplanets from their parent stars, and
• surface characteristics of the planets.
Introduction
Humans have always wondered if life exists elsewhere in the universe. Such 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 heating of the planet by its parent
star.
When a planet absorbs light from its parent star, 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 Input
The power (i.e., energy per unit time) the planet absorbs is related to the amount of starlight 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 starlight is proportional to the star's luminosity, L. A brighter star would put out
more light, so more starlight will reach the planet. The amount of starlight at the planet's distance,
d, from its star 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 INPUT ∝ L/d2 × PLANET CROSS-SECTION AREA × % ABSORBED
Energy Output
The energy given off by a planet can be estimated by using the blackbody radiation concepts. 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 the Stephan-Boltzman Law. This says that the
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
total energy given off from the planet's surface is proportional to the surface area (note: "surface
area" = 4 π "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. 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 OUTPUT ∝ T4 × PLANET SURFACE AREA × % TRANSMITTED
Note 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.
Equating power in and power out, leads us to the following relationship for the temperature of the
planet:
ENERGY INPUT = ENERGY OUTPUT
L/d2 × PLANET CROSS-SECTION AREA × % ABSORBED
∝
4
T × PLANET SURFACE AREA × % TRANSMITTED
Now, the planet's area cancels out on both sides, leaving:
T4 ∝ L/d2 × (% ABSORBED / % TRANSMITTED)
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
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
exoplanets 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 exoplanets. 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 exoplanets
(T / TEARTH)4 = (L / LSUN) × (dEARTH / d)2
where the planetary characteristics (the %'s) have been cancelled out because we assumed they
were the same for the exoplanets 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
luminosity 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. The planet and its parent 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 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,
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
d3 = P2 × M
where d, the average distance between the star and the planet (same as the semi-major axis), in
AU. The orbital period P is in years, and M, the mass of parent star, in units of the sun’s mass!
Kepler's Law has been written in the form above to make unit conversions easier; make sure you
use the correct units in this equation!
Instructions
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. Use linear interpolation on the data in Table 2 as needed.
The observed values of velocity (derived from Doppler shift) versus time for three stars harbouring
an exoplanet are given in Figures 1 to 3. Find the periods of the planets from these figures and
record this in Table 1. Use Kepler's Third Law to find the distance to the planet and record this in
Table 1. And then use the equations derived above to find the temperature of the planet; record this
in Table 1.
If you look at Figures 1 to 3 of velocity versus time for the planets, they also have a quantity called
'M sin(i)' which is the mass of the orbiting planet (the sin(i) factor accounts for the fact it is
impossible to tell how much the star-planet system is tipped from our point of view). Now you can
answer the questions in the following worksheet.
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
Exoplanet Lab – Worksheet
Data
Figure 1: Radial velocity curve for 70 Virginis.
Figure 2: Radial velocity curve for 47 Ursae Majoris.
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
Figure 3: Radial velocity curve for 51 Pegasi.
Star
Spectral Type of star
51 Pegasi
G2
47 Ursa Major
G0
70 Virginis
G4
Luminosity (Lsun)
Mass of star (Msun)
Period of planet (yr)
distance (AU)
T of planet
Liquid Water?
Table 1 - Exoplanet Data
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
Questions
1.
Is the typical mass of these three extrasolar planets greater than the mass of the Earth, or
less?
2.
Judging from their mass, what kind of planet are these extrasolar planets likely to be
(terrestrial/rocky or jovian/gas giant)?
3.
Are they likely to have a large atmosphere? Explain.
4.
When you consider the mass of the planets and the distance they are orbiting their stars, how
are these other solar systems different from our own?
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html
5.
While some of the extrasolar planets discovered so far are in the right place in their solar
system to have liquid water, many are gas giants. Do you think it would be possible for life
to develop on a gas giant? Why/why not?
6.
Could life exist in other solar systems if the planets were terrestrial planets, and not gas
giants? What factors not already discussed here do you think might impede the formation of
life in other solar systems?
*Adapted from the Univ. of Michigan Astron. Dept. lab exercise: https://dept.astro.lsa.umich.edu/ugactivities/Labs/extrasolar_planets/pn_intro.html