TGEO 110 ‘Search for Life beyond the Earth’
Spring 2012 (TTh at 2:45-4:05 PM; HU 112)
Professor John W. Delano
Mass of Sun = 1.99 x 1030 kilograms
Radius of the Sun = 6.97 x 108 meters
Mass of Earth = 5.98 x 1024 kilograms
1 Earth-year = 365.25 days = 8766 hours
Average distance from center of Sun to center of Earth = 1.49 x 1011 meters
Circumference of a circle = 2r where r = radius of circle
I. Using the concepts discussed in class, and the information provided above, determine the
following three parts of this assignment: (a) Find the distance in meters that the center of mass
of the Earth + Sun system is located from the center of the Sun; (b) Find the speed in
meters/second that the Sun wobbles in its orbit around the center of mass due only to the Earth;
and (c) If the minimum speed of a star’s wobble that can be detected by current technology is
1 meter/second, can Earth-like planets be detected around other stars at the present time? (yes
or no)
Part (i): Determine the location of the center of
mass of the Earth + Sun system.
[(weight)*(distance)]Sun = [(weight)*(distance)]Earth
(1.99 x 1030 kg)*(DS) = (5.98 x 1024 kg)*(DE)
DE / DS = (1.99 x 1030 kg) / (5.98 x 1024 kg)
DE / DS = 3.33 x 105
Therefore, … DE = (3.33 x 105) DS
DS + DE = 1.49 x 1011 meters
{(3.33 x 105) DS} + DS = 1.49 x 1011 meters
DS = 4.47 x 105 meters from center of Sun
Note: Radius of the Sun = 6.97 x 108 meters
Part (ii): Determine the speed of the Sun’s wobble
around the center of mass.
Circumference of ‘wobble’ orbit = 2R
where R = DS
circumference of Sun’s wobble = 2.81 x 106 meters
The Sun makes 1 ‘wobble’ orbit in 1 Earth-year,
which is 3.156 x 107 seconds
Velocity = distance / time
Velocity = (2.81 x 106 meters) / (3.156 x 107 sec)
Velocity = 0.089 meter/sec
Part (iii): ‘No’. Since the Sun’s wobble due to the
Earth has a velocity that is < 1 meter/second,
current technology does not allow us (yet) to be
able to detect Earth-sized planets orbiting other
stars.
II. As the Earth+Moon system orbits the Sun in 1 year (365.25 days), the Moon orbits the
Earth with an average period of 27.3 days. The mass of the Earth is 5.98 x 1024 kilograms. The
mass of the Moon is 7.34 x 1022 kilograms. The average distance between the center of the
Earth and the center of the Moon is 3.80 x 108 meters. The average radius of the Earth is 6.37 x
106 meters.
(i)
Determine the location of the center of mass of the Earth+Moon system with respect
to its distance (in kilometers) from the center of the Earth.
(ii) Determine the speed (in meters/second) that the Earth wobbles around the
Earth+Moon center of mass.
Part (i): Determine the location of the center
of mass relative (a) the center of the Earth.
(Earth’s mass)(DE) = (Moon’s mass)(DM)
(5.98 x 1024 kg)(DE) = (7.34 x 1022 kg)(DM)
(5.98 x 1024 kg) / (7.34 x 1022 kg) = (DM/DE)
81.47 = (DM/DE)
Therefore: 81.47(DE) = DM
Since DE + DM = 3.80 x 108 meters
then
DE + (81.47 DE) = 3.80 x 108 meters
82.47 (DE) = 3.80 x 108 meters
DE = 4.61 x 106 meters from center of Earth
Note: Earth’s radius = 6.37 x 106 meters
Part (ii): What is the speed (meters/second) of
the Earth’s wobble?
Speed = distance / time
Period of Earth’s wobble = 27.3 days
= (27.3 days) (24 hr/day) (60 min/hr) (60 sec/min)
= 2.36 x 106 seconds
Distance of 1 Earth wobble = 2 (DE)
= 2 (4.61 x 106 meters)
= 2.90 x 107 meters
Speed = (2.90 x 107 meters) / (2.36 x 106 seconds)
Speed = 12.3 meters/second