# PHS1110 Columbia Southern Calculate Escape Velocities Project InstructionsConceptual Experiment For this activity, you are going to calculate escape veloci

PHS1110 Columbia Southern Calculate Escape Velocities Project InstructionsConceptual 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. 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

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, mv2/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

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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=

(6.6710-11 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= 1.501011 m. The mass of the earth is m = 5.9810 24 kg and the

mass of the Sun is M=1.991030 kg.

Solution: From F=GmM/r2= 6.6710-11 x 5.981024 x 1.991030 / (1.50 x1011)2 = 3.51025 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

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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

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