Historical Geology: Planet Orbiting

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Lab 13 problems.docx 
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Lab 13: 51 Pegasi
Summary This investigation details how scientists discovered a planet orbiting another star and
compares the results of the discovery with planets in our solar system.
Materials
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 red-shifted) 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 companion.
Observations
TABLE v (m/s)
1: 51
Pegasi
Radial
Velocity
Data
Day
0.6
-20.2
0.7
-8.1
0.8
5.6
1.6
56.4
1.7
66.8
3.6
-35.1
3.7
-42.6
4.6
-33.5
Day
v (m/s)
Day
v (m/s)
Day
v (m/s)
4.7
4.8
5.6
5.7
5.8
6.6
6.7
7.7
-27.5
-22.7
45.3
47.6
56.2
65.3
62.5
-22.6
7.8
8.6
8.7
8.8
9.6
9.7
9.8
10.6
-31.7
-44.1
-37.1
-35.3
25.1
35.7
41.2
61.3
10.7
10.8
11.7
11.8
12.6
12.7
13.6
13.7
56.9
51
-2.5
-4.6
-38.5
-48.7
2.7
17.6
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.
Procedure:
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 missing.
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.)
Worksheet
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 velocities.)
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:
a. the orbit of the planet is circular (e = 0)
b. the mass of the star is 1 solar mass
c. the mass of the planet is much, much less that of the star
d. we are viewing the system nearly edge on
e. we express everything in terms of the mass and period of Jupiter
We makethese 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 semi-major 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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