Astrobiology in the Classroom

 

NASA – CERES Project –http://btc.montana.edu/ceres

Montana State University

Preliminary Edition

 

 

 

“A Case of the Wobbles: Finding Extra-Solar Planets

 

 

 

 

 

 

 

 

Students plot and analyze NASA data to determine the period of an invisible planet orbiting a wobbling star. In this three-part guided inquiry activity, students first explore the motion of a two-body system around a center of mass to better understand how extra-solar planets are discovered.  Students are shown a drawing depicting the orbital motion of the Sun and Jupiter and answer reflective-questions designed to illustrate that the Sun is not a stationary object, but that all stars move or wobble when orbited by a large planet.  Student then tackle questions about the magnitude and direction of the velocity of both the Sun and Jupiter.  In the second part, students view movies of athletes competing in the Hammer Throw.  Students discover that the hammer and thrower both move about a common center of mass.  They find that the center of mass is determined by considering the positions and masses of the two bodies.    Finally, students use actual scientific data from recent discoveries of extra-solar planets.  By plotting the central star’s velocity toward or away from an observer versus time, students determine the period of an invisible planet’s orbit.

 


 

Part I – Exploration: The Motion of the Sun

 

The four drawings below show the positions of the Sun and Jupiter at four different times during a single orbit.  Since the Sun and Jupiter account for nearly all the mass of our solar system we will model our solar system as a two-body problem involving only the Sun and Jupiter. 

 

March 1988

 

March 1985

 
 


 

 

 

 

Observer

 

Observer

 

March 1991

 

March 1994

 
 

 

 

 

 

 

 

 


 

 

 

Observer

 

Observer

 
 

 

 

 

 

 

 


A.    Does the Sun appear to always remain in the same position?  If not, describe its motion.

 

 

 

B.    Estimate the time (in Earth years) for one orbit of the Sun (this time is known as the orbital period)?  How does this time compare to the orbital period of Jupiter?

C.    In each diagram, find and label the locations of the center of the Sun’s orbit and the center of Jupiter’s orbit.  If these two positions coincide with each other, this position is the Center of Mass (CM) for the system.

 

D.   On each drawing of the Sun and Jupiter system, sketch a vector to represent the velocity of each orbiting object. What does the direction of the vector represent? What does the length of the vector represent? How does the length of the vector you drew for Jupiter compare to the length of the vector you for near the Sun?

 

 

 

 

 

 

E.    When studying motion it is useful to consider the object’s velocity as being made of two components (or parts.)  The component of velocity that is directed toward (negative) or away (positive) from the observer’s line of sight is known as the radial velocity.  The component of velocity that is directed perpendicular to the observer’s line of sight is known as the tangential velocity.  Imagine that you are the observer shown in each of the drawings.

 

                 i.     At which date(s) would the Sun appear to be moving with a radial velocity?

 

 

                  ii.     At which date(s) would the Sun appear to be moving with a tangential velocity?

 

 

                    iii.     At which date(s) would Jupiter appear to be moving with a radial velocity?

 

 

                   iv.     At which date(s) would Jupiter appear to be moving with a tangential velocity?

 

 

F.    What would you see in September 1992 from the position of the observer? Draw a sketch like those shown on the previous page. Include velocity vectors for the Sun and Jupiter.

 

 

 

 

 

 

 

 

 

 

                     i.     From the location of the observer, does the Sun appear to be moving with a radial velocity, or with a tangential velocity, or both?  Explain your reasoning.

 

 

 

 

                       ii.     Does Jupiter appear to be moving with a radial velocity, or with a tangential velocity, or both? Explain your reasoning.  Explain your reasoning.

 

 

 

 

G.   Imagine that you are located at the observer location shown in each drawing and that there is a satellite located near the Sun on the side closest to you.  If the satellite was sending out a distress signal: 

 

            i.         At which date would the message arrive during the shortest interval of time?

 

 

              ii.         At which date would the signal take the greatest amount of time?

 

 

               iii.         During what time interval would the signal have a Doppler shift to a longer wavelength?

 

 

               iv.         During what time interval would the signal have a Doppler shift to a shorter wavelength?

 


Part II – Concept Introduction:  The Hammer Throw

 

A.    Watch the video of the hammer throw (top view) available on the Internet at http://btc.montana.edu/ceres/astrobiology/movies/ht_top.mpg

(Note: your teacher may have already downloaded this movie for you)

 

B.    Watch the video of the hammer throw (side view) available at http://btc.montana.edu/ceres/astrobiology/movies/ht_side.mpg 

(Note: your teacher may have already downloaded this movie for you)

 

C.    Is the thrower able to stay in one location as the hammer is swung around?  What object or force is causing the thrower to move from side to side as he spins?

 

 

 

D.   Draw four overhead sketches (like the drawings provided in Part I. of the Sun and Jupiter) of the hammer and thrower for four different instances during one complete orbit.  Include dotted circles with each sketch to illustrate the entire path of the orbit for both the hammer and thrower (as was shown in the drawings provided in Part I.)

 

 

 

 

 

 

 

 

 

 

 

 

E.    In each drawing mark the center of the orbit for each object. 

 

F.    Should the centers of each object's orbit lie at the same location?  If so, what is the name given to this location?  If not why not?

 

 

 

 

G.   Is the location you marked closer to the thrower, the hammer, or halfway between the two?  Explain your reasoning.

 

 


 

H.   If a more massive hammer is used, does the location of the center of mass move closer to or farther from the hammer?  Does the location move closer to or farther from the thrower?  Draw a sketch to illustrate your answer.

 

 

 

 

 

 

 

 

I.      Compare the size of the orbital paths (circles) that you drew in part D to those that you drew in part H.  Did the size of each object's orbit increase, decrease or the stay the same? 

 

 

 

J.     If the mass of the hammer and thrower are equal how does the distance from the center of mass to each object compare?  Draw a sketch to illustrate your answer.

 

 

 

 

 

 

 

 

K.   The equation provided below is used to calculate the location of the center of mass for a two-body system.  For the thrower & hammer system shown let the mass of the thrower m1 = 120 kg, the mass of the hammer m2 = 40 kg.  The distance from the origin to the thrower is represented by the symbol r1 and the distance from the origin to the hammer is represented by the symbol r2.  The values of r1 and r2 can be negative. The hammer and thrower are 1.5 m apart.  

 

 

 

 

 


i.      Calculate the location of the center of mass for a coordinate system with the origin chosen to be at the midpoint between the hammer and thrower.  Show your work and mark this location in the drawing above.

 

 

 

ii.     Calculate the location of the center of mass for a coordinate system with the origin chosen to be at the location of the hammer.  Show your work and mark this location in the drawing above.

 

 

 

iii.   Calculate the location of the center of mass for a coordinate system with the origin chosen to be at the location of the thrower.  Show your work and mark this location in the drawing above.

 

 

 

L.    Does the actual location of the center of mass change with the different choices of  coordinate systems?

 

 

M.  The top-view path for one orbit of the thrower is shown below, the location of the hammer has not been drawn.  Using the length and size of the rope and hammer shown below mark the location of the hammer for each position of the thrower. 

 

 

 

 


 

2

 

Rope and hammer used in the throw.

 
A

 

 

 

 

 

B

 

 

 

N.   With each thrower location, are there any other locations that the hammer could have been drawn at other than the one you have shown?  Why or why not?

 

 

 

O.   At each location 1-4, draw a vector to represent the velocity of the thrower.

 




 

P.    If you were standing at point A, for which position(s) of the thrower would you measure the radial velocity to be:

i.    positive

ii.   negative

iii.  zero

 

Explain your reasoning.

 

 

 

Q.   Carefully sketch a radial velocity vs. time graph for the thrower as seen by an observer at point A.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R.    How would you change the above graph if it were made by an observer at point B?  Explain your reasoning.

 

 


Part III – Concept Application: Discovering New Planets

 

In this activity, you will examine real data gathered by astronomers in their pursuit to find planets orbiting distant stars. 

 

A.  The tables below provide information about the motion of two stars (51 Pegasi, and Tau Boo).*  Based on the data provided in these tables, do you believe there is a companion planet in orbit about these stars?  Explain your reasoning.

Text Box: Tau Boo 

Time (Days)	Radial Velocity (m/s)
0	-550
0.1655	-600
0.4965	-450
0.662	-300
0.993	0
1.1585	100
1.4233	300
1.7543	350
1.986	300
2.1515	200
2.317	50
2.5487	-50
3.31	-550
3.4755	-600
3.8065	-450

 

Text Box: 51 Pegasi

Time (Days)	Radial Velocity (m/s)
10	55
11	-30
12	-32
13	32
14	70
15	-20
16	-42
17	20
18	52
19	48
20	-45
21	0
22	70
23	25
24	-19
25	-20
27	70

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B.    Describe how the motion of the star differs when it has a positive versus a negative velocity. 

 

 

 

C.    How do we detect that the stars are moving with a radial velocity?  Hint consider your answers to activity #1 question “G”.

 

 

 

 

* This data was inferred form the actual velocity versus time graphs, which can be found at: http://cannon.sfsu.edu/~gmarcy/planetsearch/doppler.html)


D.  Use the grid below to construct a radial velocity versus time graph for 51 Pegasi.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


D.  Use the grid below to construct a radial velocity versus time graph for Tau Boo.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E.   Describe the similarities and differences in your graphs.

 

 

 

 

 

F.   Based on your graphs, would you infer that the stars 51 Pegasi and Tau Boo are in orbit with a companion planet?  Explain your reasoning.

 

 

 

 


G.  How long would it take the companion planet to complete one full orbit for each of these stars?  Explain your reasoning.

 

 

 

H.   Below are the velocity versus time graphs for two stars (47 Ursae Majoris and HR 3522)* that might have a companion planets in orbit about them.  For each star state whether or not you think the star has a companion planet and if so estimate the orbital period of the planet.  If not, explain why not.

 

i.    47 Ursae Majoris

 

 

 

ii.   HR 3522

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


* The original versions of these graphs can be found at: http://cannon.sfsu.edu/~gmarcy/planetsearch/doppler.html


 

I.     

55 Rho cnc

 
At what time(s) was each star moving towards earth with the greatest velocity? 

 

i.    47 Ursae Majoris                                      ii.        55 Rho cnc

 

 

 

 

J.     At what time(s) was each companion planet moving towards earth with the greatest velocity? 

 

i.    47 Ursae Majoris                                      ii.        55 Rho cnc

 

 

 

 

K.   Was the velocity of either star ever zero?  If so when, and if not why not? 

 

i.    47 Ursae Majoris                                      ii.        55 Rho cnc

 

 

 

 

L.    What was the maximum value of the tangential velocity of each star?  Explain your reasoning.

 

 

M.  Like our Sun, both  47 Ursae Majoris, and 55 Rho cnc, are “G” type stars.  Therefore to a good approximation we know that each of these stars have a mass of approximately 2 x 1030 kg.  Scientists have estimated that the companion planet orbiting 47 Ursae Majoris has a mass of 2.6 Jupiter masses (mass of Jupiter is 1.9 x 1027 kg) and is 2.11 AU’s ( 1 Au = 1.5x1011m) away from the star.  They have also estimated that the companion planet orbiting 55 Rho cnc has a mass of .84 Jupiter masses (mass of Jupiter is 1.9 x 1027 kg) and is .11 AU’s ( 1 Au = 1.5x1011m) from the star.

 

i.      Find the location of the center of mass for the 47 Ursae Majoris and planet system.  Draw a sketch of the system below and label the planet, star and location of the center of mass. 

 

 

 

 

 

 

 

 

 

 

 

 

ii.     Find the location of the center of mass for the 55 Rho cnc and planet system.  Draw a sketch of the system below and label the planet, star and location of the center of mass.