Lab: Finding Exoplanets, geology homework help


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Lab 13: 51 Pegasi
This investigation details how scientists discovered a
planet orbiting another star and compares the results of
the discovery with planets in our solar system.

Graph Paper
Scientific Calculator
Background and Theory
In just the past few years, astronomers have announced discoveries of at least 30
planets orbiting nearby stars. These discoveries seem to finally answer the question of
whether or not our solar system is unique. We should note, however, that when
astronomers state that they have discovered a new planet, what they are really saying is
that their data can best be interpreted as a planet orbiting a star. One cannot “prove”
that these other planets exist (short of actually going there to explore!); one can only
state that, until the hypothesis is disproved, a planet orbiting the star best explains the
observations. We cannot see these planets. We can only measure indirectly the
influence each one has on its parent star as the star and planet orbit their common
center of mass. The planet makes the star “wobble.”
We enter this realm of discovery by working with actual data from observations of the
star 51 Pegasi (51 Peg) made at the Lick Observatory in California. These data are the
measurements of the Doppler shift of the wavelengths of the absorption lines seen in the
spectra of 51 Peg. Table 1 lists the measured radial velocities (RV) as a function of time
(recorded in days). As you can see, the radial velocities are sometimes positive and
sometimes negative indicating that sometimes the star is receding from (the light is redshifted) and sometimes approaching (the light is blue-shifted) our frame of reference.
This wobble of the star was the first indication that the star 51 Peg had an invisible
TABLE 1: 51 Pegasi Radial Velocity Data
v (m/s)
v (m/s)
v (m/s)
v (m/s)
Note: The days of the observations in this table are expressed in the number of
days, or fraction thereof, from when the astronomer first started observing. That
is, the dome of the telescope was first opened at Day = 0.
Table 1 lists the observed radial velocities. These were obtained by measuring the
Doppler shift for the absorption lines using the formula:
Solving for the radial velocity v of the star:
Here, c is the speed of light, is the laboratory wavelength of the absorption line being
measured, and
is the difference between the measured wavelength of the line and
the laboratory value.
1. Plot the 32 data points on graph paper, setting up your scale and labels. Use the
observed radial velocities (in m/s) versus the day of the observation.
2. Draw a smooth curve (do not simply connect-the-dots) through the data. The
curve is a sine curve (ask if you don’t know) and thus will always reach the same
maximum and minimum values and have the same “number of days” between
each “peak” and “valley”. You should interpolate between data where points are
3. Thought question: Why are there data missing? Why are there sizable gaps in
the data? (Hint, some gaps are a little over 1/2 day long and these are
observations from the ground.)
1. A period is defined as one complete cycle; that is, where the radial velocities
return to the same position on the curve (but at a later time).
How many cycles did the star go through during the 14 days of observations?
Number of cycles = ___________
2. What is the period, P, in days?
Period = ___________ days
3. What is P in years?
P = _____________ years
4. What is the uncertainty in your determination of the period? That is, by how many
days or fractions of a day could your value be wrong?
Uncertainty = ___________ days
5. What is the amplitude, K? (Take 1/2 of the value of the full range of the
K = ___________ m/s
6. How accurate is your determination of this value?
Uncertainty = __________ m/s
7. We will make some simplifying assumptions for this new planetary system:
the orbit of the planet is circular (e = 0)
the mass of the star is 1 solar mass
the mass of the planet is much, much less that of the star
we are viewing the system nearly edge on
we express everything in terms of the mass and period of Jupiter
We make these assumptions to simplify the equations we have to use for
determining the mass of the planet. The equation we must use is:
P should be expressed in years (or fraction of a year), and K in m/s. Twelve
years is the approximate orbital period for Jupiter and 13 m/s is the magnitude of
the “wobble” of the Sun due to Jupiter’s gravitational pull. Not all calculators will
take the cube root of a number. Get help if yours does not. Put in your values for
P and K and calculate the mass of this new planet in terms of the mass of
Jupiter. That is, your calculations will give the mass of the planet as some factor
times the mass of Jupiter (for example: Mplanet = 4 MJupiter). Show all work.
8. Assume that the parent star is 1 solar mass, and that the planet is much less
massive than the star. We can then calculate the distance this planet is away
from its star, in astronomical units (AU’s) by using Kepler’s third law:
Again, P is expressed in years (or fraction of a year), and a represents the semimajor axis in AU’s. Solve for a:
a = __________ AU
9. Compare this planet to those in our solar system. For example, Mercury is 0.4
AU from the Sun; Venus, 0.7 AU; Earth, 1.0 AU; Mars, 1.5 AU; Jupiter, 5.2 AU.
Jupiter is more massive than all the rest of the matter in the solar system
combined, excluding the Sun.
What is unusual about this new planet?
10. Science is based upon the ability to predict outcomes. However, nothing
prepared astronomers for the characteristics of this “new” solar system. Why was
it such a surprise?
11. If this actually is a planet, is it possibly hospitable to life? Explain.
12. Name your new planet — a privilege you would have if you really did discover a
new planet!

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