PHS1110 Columbia Southern Exoplanets Escape Velocities Calculation Assignment

Conceptual Experiment

For this activity, you are going to calculate escape velocities for several exoplanets and compare them with our major planets.

In order to find escape velocities for the given exoplanets, enter the appropriate values of their masses and radii in unit of Earth’s mass and Earth’s radius in the provided template. Then, the escape velocity and gravitational acceleration will be automatically evaluated. In the case of our major planets, insert data from Table 10.1 on p. 199 in the textbook. After completing the table, answer the questions directly on the worksheet. You will save and upload your work on the provided template and submit it when you are complete.

The Table 10.1 from textbook is attached along with assignment template worksheet, assignment instruction and study guide.

Escape Velocities on Exoplanets
Are we alone? Is the solar system unique in the Universe? No. It is just difficult to find planets because they are so tiny and
dark compared to stars. Maybe direct observation is impossible even if the planet is larger than Jupiter because of the
brightness of stars. However, there are some indirect methods to find them, and so far about 2,000 planetary systems have
been found. In 1992 for the first time, two planets circling around Pulsar PSR 1257+12 were founded by radio astronomers
using the pulsar timing method. In 1995, a planet orbiting around a main sequence star like the sun, 51 Pegasi, was
discovered using radial velocity method. After that, many extra solar planets (exoplanets) were discovered using numerous
indirect methods. For more further information, you may visit the following websites:
http://exoplanetarchive.ipac.caltech.edu/
http://planetquest.jpl.nasa.gov/
http://planetquest.jpl.nasa.gov/news/239#
For this activity, we are going to calculate escape velocities for several exoplanets and compare them with our major planets.
Escape velocity, Ve, is defined to be the minimum velocity an object must have in order to escape the gravitational field of
planet, that is, escape the planet without ever falling back. It can be evaluated by
ve =
2 GM
R
=
2 gR
where M is the mass of the planet, G is the gravitational constant, g is acceleration of gravity on the planet’s surface,
the radius of the planet.
The selected exoplanets with some physical properties are as follows.
Kepler- 452b is located about 1,400 light years away from earth. Its size is 1.6 times of Earth’s radius and it has 5 times
Earth’s mass.
51 Pegasi b is about 41 light years from Earth. Its mass is about half of that of Jupiter. Its size is about twice of that of
Jupiter’s mass is 318 times Earth’s mass and Jupiter’s size is about 11 times Earth’s size.
Kepler-78b is located about 400 light-years from Earth. Its mass is double of Earth’s mass and its size is 1.2 times of Earth’s
radius.
The distance from “Super-Earth” exoplanet, OGLE-2005-BLG-390 Lb, is about 22,000 light years.
of Earth. It is five times heavier than that of Earth.
The distance between WASP-18b and Earth is about 325 light years. The mass of WASP-18b is 10 times of Jupiter’s mass,
that is, about 3,180 times the mass of Earth. Its size is 11 times bigger than that of Earth’s radius.
Look at the provided table below. The first column is the name of planet, the second column is the mass, the third column is
the radius, the fourth column is the escape velocity, Ve, and the fifth column is gravitational acceleration, g.
escape velocities for the given exoplanets, enter the appropriate values of their masses and radii in unit of Earth’s mass and
Earth’s radius in the table below. Then, the escape velocity and gravitational acceleration will be automatically calculated.
the case of our major planets, insert data from Table 10.1 on p. 199 in the textbook. Then, you will get them, too. After
completing the table, answer the questions.
Microsoft excel software is required to use the table below for automated calculation. However, if you do not have this,
you can use your own calculator. It should be no problem to answer the questions.
1. Fill out Mass and Radius columns below using above information.
Exoplanet name
51 Pegasi b
Kepler -78b
Kepler-452b
Mass [ME] Radius [RE]
Ve[km/s]
g [m/s/s]
WASP-18b
OGLE-2005-BLG-390 Lb
Our Planets
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
2.Which planet (including both exoplanets and our major planets) is the most difficult to escape?
3.Which planet (including both exoplanets and our major planets) has the largest gravitational field, and which planet has th
gravitational field?
4.Which exoplanet is most like the earth? Justify your answer.
Your response should be at least 50 words in length.
5.Which factor affects the escape speed? Mass and/or radius?
Your response should be at least 50 words in length.
6. If one of the planets becomes a black hole, what would the escape speed be?
Your response should be at least 50 words in length.
ult to find planets because they are so tiny and
lanet is larger than Jupiter because of the
and so far about 2,000 planetary systems have
R 1257+12 were founded by radio astronomers
quence star like the sun, 51 Pegasi, was
(exoplanets) were discovered using numerous
lanets and compare them with our major planets.
ave in order to escape the gravitational field of the
eration of gravity on the planet’s surface, and R is
.6 times of Earth’s radius and it has 5 times
of Jupiter. Its size is about twice of that of Jupiter.
Earth’s mass and its size is 1.2 times of Earth’s
out 22,000 light years. Its size is about half of that
18b is 10 times of Jupiter’s mass,
that of Earth’s radius.
second column is the mass, the third column is
is gravitational acceleration, g. In order to find the
eir masses and radii in unit of Earth’s mass and
acceleration will be automatically calculated. In
extbook. Then, you will get them, too. After
However, if you do not have this,
ost difficult to escape?
gravitational field, and which planet has the smallest
•
Conceptual Experiment
For this activity, you are going to calculate escape velocities for several exoplanets and
compare them with our major planets.
In order to find escape velocities for the given exoplanets, enter the appropriate values of
their masses and radii in unit of Earth’s mass and Earth’s radius in the provided template.
Then, the escape velocity and gravitational acceleration will be automatically evaluated. In
the case of our major planets, insert data from Table 10.1 on p. 199 in the textbook. After
completing the table, answer the questions directly on the worksheet. You will save and
upload your work on the provided template and submit it when you are complete.
UNIT IV STUDY GUIDE
Gravity and Orbital Motion
Course Learning Outcomes for Unit IV
Upon completion of this unit, students should be able to:
3. Explain Newton’s laws of motion at work in common phenomena.
3.1 Illustrate the relation of the universal law of gravitation to Newton’s second law.
3.2 Distinguish between gravitational acceleration (g) and gravitational constant (G).
3.3 Evaluate gravitational field strength when mass and radius of an object are given.
4. Explain the concepts and applications of momentum, work, mechanical energy, and general relativity.
4.1 Apply total mechanical energy conservation for orbital motion.
4.2 Calculate escape velocity when gravitational potential energy is balanced with kinetic energy.
4.3 Describe the escape velocity in a black hole, a consequence of Einstein’s general relativity.
Reading Assignment
Chapter 9: Gravity
Chapter 10: Projectile and Satellite Motion
Unit Lesson
Projectile Motion
When an object moves with a curved path near the earth’s surface under the influence of gravity, its motion is
called projectile motion. For example, look at Figures 10.6 and 10.8 on pages 185 to 186 in the textbook.
If we ignore air resistance, the horizontal motion of the projectile does not slow down; its velocity is constant.
In other words, the horizontal component of the acceleration is zero. However, the vertical component of the
velocity is not constant, but changes. In addition, the vertical component of the acceleration is downward
acceleration, gravitational acceleration, (g).
Weightlessness and Free Fall
Suppose you are in an elevator. If the elevator is not accelerating, your weight (W) is just your mass (m) times
the gravitational acceleration (g). In fact, two forces are acting on you; the weight (W) and the normal force
(F). According to Newton’s second law, in the vertical direction, ma=F-W=F-mg. That is, normal force
F=m(g+a). Here, g is positive, but a may be either positive for upward acceleration or negative for downward
acceleration of the elevator. If the elevator is in upward motion, apparent weight (or normal force) is greater
than your true weight. On the other hand, if the elevator is in downward motion, the apparent weight is smaller
than your true weight. In a special case, when the acceleration is equal to g, that is, a=-g, or free fall, the
apparent weight becomes zero: weightless. Please look at Figure 9.9 on p.166 in the textbook for an example
of this. The same phenomena occur when an object is circling around the earth. The orbiting satellite, which
accelerates toward the center of the earth, is also in free fall. See Figure 9.10 on p.167 in the textbook.
Over a long period of time, the weightlessness is harmful for humans, and thus, a rotating space station in a
wheel shape is provided to create artificial gravity. It is balanced with the centripetal force, mv 2/r, of the
system. That is mg=mv2/r. Here, m is the mass of an astronaut, r is the distance from axis to the surface of
the station, and v is the rotating speed. For instance, if r is given 1 km, then v=(rg)1/2= 100 m/s.
PHS 1110, Principles of Classical Physical Science
1
Newton’s Law of Universal Gravitation
Newton speculated about the highest reachable point by the force of gravity on the earth. He realized that
there is a limit and concluded that the orbital motion of the moon around the earth is maintained by the
gravitational force (Hewitt, 2015). Suppose you throw a stone horizontally from a high place (See Figure
10.16 on p. 190 in the textbook). The stone falls to the ground because of gravity. However, if you throw the
stone with great speed, it will move further and further away from where you are standing before falling to the
ground. When the speed is great enough, the stone will eventually circle around the earth. This is the
projectile motion, where the projectile falls in the gravitational field but never touches the ground. This logical
consideration can be applied to explain the orbital motion of the moon. Newton concluded that the moon is
falling in its pathway around the Earth because of the gravitational acceleration.
Newton extended the above idea to any two objects in the universe in order to explain the interaction between
them. Newton’s law of universal gravitation postulates that there is an attractive force between the two objects
(Hewitt, 2015). The force between two objects in the universe is proportional to the product of two masses m
and M and is inversely proportional to the square of distance r between two objects; F=GmM/r2 , where G=
(6.6710-11 N m2/kg2) is the universal gravitational constant. This is the case when the gravitational
acceleration (a) is equal to g in the second law of Newton; a=g, and thus, g=GM/r2. The constant, G was
measured by Cavendish 100 years after Newton announced his theory. It was not an easy task because of
the extremely small value of gravitation attraction. The detailed story is in Section 9.2 on pp. 163–164 in the
textbook.
Example: What is the magnitude of the gravitational force between the sun and the earth? The distance
between the sun and the earth is 1AU= 1.501011 m. The mass of the earth is m = 5.9810 24 kg and the
mass of the Sun is M=1.991030 kg.
Solution: From F=GmM/r2= 6.6710-11 x 5.981024 x 1.991030 / (1.50 x1011)2 = 3.51022 N
Kepler’s Three Empirical Laws for Planetary Motion
Johannes Kepler (1571 – 1630) was a German astronomer and had an endless enthusiasm for researching
the solar system. It took him more than 20 years to realize through his calculations the exact shape of the
planets’ orbitals. He tested many different kinds of models using his teacher Tyco Brache’s enormous data
set. Brache had accumulated very exact planetary data without even the use of telescopes. Kepler
established three important empirical laws of planetary motion: the law of elliptical orbit, the law of areas, and
the law of the relation between period and distance. This is what he used to describe and understand the
motion of the Solar System (Zeilik & Smith, 1987).
Mechanical motion of our solar system obeys gravitational law, and planets are orbiting around the sun, which
is the heaviest mass in the solar system. The orbital shape is not circular, but elliptical. Some comets have
parabolic or hyperbolic orbits. These well-known mechanics were not easily discovered. Since the ancient
times, the sky was considered a realm of gods, so perfectness was assumed. The notion that the orbits of
planets should be a perfect circle was widely accepted, and no scholar would be able to prove otherwise to
the people, even Kepler. For these reasons, it took a long time to accurately describe planetary motion in our
solar system. The famous three empirical laws of planetary motion describe the motion of the solar system as
follows:

First Law—The law of ellipses: The orbit of each planet is an ellipse with the sun at one foci. The
shape of a planet’s orbit is an ellipse.

Second Law—The law of areas: The radius vector to a planet sweeps out equal areas in equal
intervals of time. When a planet is closer to the sun, it revolves faster, and, on the other hand, when a
planet is farther away from the sun, it revolves slower.
PHS 1110, Principles of Classical Physical Science
2

Third Law—The law of harmony: The squares of the sidereal periods of the planets are proportional
to the cubes of the semi-major axes (mean radii) of their orbits. Here, the sidereal period is the time it
takes the planet to complete one orbit of the Sun with respect to the stars (Zeilik & Smith, 1987).
Thus, Kepler’s laws and Newton’s laws taken together imply that a force holds a planet in its orbit by
continuously changing the planet’s velocity, so that it follows an elliptical path. The force is directed toward the
sun from the planet and is proportional to the product of the masses of the sun and the planets. Also, the
force is inversely proportional to the square of the planet-sun separation. This is precisely the form of the
gravitational force postulated by Newton. Newton’s laws of motion, with a gravitational force used in the
second law, imply Kepler’s laws, and the rest of the planets obey the same laws of motion as objects on the
surface of the earth.
Conic Sections and Gravitational Orbits
Hypatia (360 – 415) visualized various shapes of geometric equations using conic sections for the first time in
Alexandria, Egypt (Larson & Edwards, 2010). Conic sections are formed when a cone is cut with a plane at
various angles. For a more detailed description, visit the website about this in the Suggested Reading section
of this unit.
There are various orbits in a gravitational system. The circular orbit is a special case of ellipse. The ellipse
can be formed when the plane intersects opposite “edges” of the cone. In the case of the parabola orbit, the
plane is parallel to one edge of the cone. On the other hand, the hyperbola orbit does not intersect opposite
edges of the cone, and the plane is not parallel to the edge. Planets in our solar system have elliptical orbits
with various eccentricities. The orbital eccentricity (e) determines the shape of orbits. If e=0 (E