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Generic Conical Orbits, Keplers Laws,Satellite

Orbits and Orbital Mechanics

- Guido Cervone

Summary

- Class Website
- Continuation of previous lecture
- Generic Conical Orbits, Keplers Laws,Satellite

Orbits and Orbital Mechanics - Video - Satellite Launch

Topics

- The Remote sensing data we are treating come from

spaceborne satellites - In this lecture we will discuss the concepts of

orbiting satellites - We will discuss
- Generic Conical Orbits
- Keplerian Laws
- Celestial Mechanics
- Satellite Orbits
- Mathematical Formalization

Approach

- Mathematics is necessary to fully understand the

problem, but can be complicated because of 3D - We will approach the problem in two ways
- Conceptual. Without any math or formulas, just

exploring the concepts - Mathematical. Formulas and equations

Earths Gravitational Pull

- The Earth's gravity pulls everything toward the

Earth. In order to orbit the Earth, the velocity

of a body must be great enough to overcome the

downward force of gravity - One important fact to remember is that orbits

within the Earth's atmosphere do not really

exist. Atmospheric friction caused by the

molecules of air (causing a frictional heating

effect) will slow any object that could try to

attain orbital velocity within the atmosphere. - In space, with virtually no atmosphere to cause

friction satellites can travel at velocities

strong enough to counteract the downward pull of

Earth's gravity - The satellite is said to orbit around the Earth

How do Satellites Orbit?

Orbits

- Orbit refers to the path of a smaller object

(secondary) around a bigger object (primary) as a

result of the combined effects of inertia and

gravity.

Conic Orbits

- The orbit can be in the shape of one of four

conic sections - Circle, Ellipse, Parabola, Hyperbola
- A conic section is the shape formed on a plane

passing through a right circular cone.

Conic Orbits II

Conic Orbits III

- Most satellite and planetary orbits are elliptical

Review of Ellipses

- For an ellipse there are two points called foci

(singular focus) such that the sum of the

distances to the foci from any point on the

ellipse is a constant. - a b constant
- The long axis of the ellipse is called the major

axis, while the short axis is called the minor

axis. - Half of the major axis is termed a semi-major

axis. - The length of a semi-major axis is often termed

the size of the ellipse. - It can be shown that the average separation of a

secondary from the primary as it goes around its

elliptical orbit is equal to the length of the

semi-major axis. - Thus, by the "radius" of an orbit one usually

means the length of the semi-major axis.

Eccentricity

- The amount of "flattening" of the ellipse is

termed the eccentricity. - A circle may be viewed as a special case of an

ellipse with zero eccentricity, while as the

ellipse becomes more flattened the eccentricity

approaches one. - Thus, all ellipses have eccentricities lying

between zero and one. - The range for the eccentricities of the different

types of orbits follows circular e 0,

elliptical 0 gt e lt 1, parabolic e 1, hyperbolic

e gt 1. - The eccentricity for ellipses is the ratio of

distance between the two foci and the length of

the major axis.

Keplers laws

- In the early 1600s, Johannes Kepler proposed

three laws of planetary motion - Kepler was able to summarize the carefully

collected data of his mentor - Tycho Brahe - with

three statements which described the motion of

planets in a sun-centered solar system - The laws are still considered an accurate

description of the motion of any planet and any

satellite

Keplers First Law

- Kepler's First Law
- The orbits of the planets are ellipses, with the

Sun at one focus of the ellipse. (generally there

is nothing at the other focus of the ellipse). - The planet then follows the ellipse in its orbit,

which means that the Earth-Sun distance is

constantly changing as the planet goes around its

orbit.

Keplers Second Law

- The line joining the planet to the Sun sweeps out

equal areas in equal times as the planet travels

around the ellipse. - Thus, a planet executes elliptical motion with

constantly changing angular speed as it moves

about its orbit. - The point of nearest approach of the planet to

the Sun is termed perihelion. The point of

greatest separation is termed aphelion. - Hence, by Kepler's second law, the planet moves

fastest when it is near perihelion and slowest

when it is near aphelion.

Keplers Third Law

- The ratio of the squares of the revolutionary

periods for two planets is equal to the ratio of

the cubes of their semi-major axes. - In this equation P represents the period of

revolution for a planet and R represents the

length of its semi-major axis. The subscripts "1"

and "2" distinguish quantities for planet 1 and 2

respectively. - Kepler's Third Law implies that the period for a

planet to orbit the Sun increases rapidly with

the radius of its orbit. Thus, we find that

Mercury, the innermost planet, takes only 88 days

to orbit the Sun but the outermost planet (Pluto)

requires 248 years to do the same.

Calculations of Keplers Third Law

- A convenient unit of measurement for periods is

in Earth years, and a convenient unit of

measurement for distances is the average

separation of the Earth from the Sun, which is

termed an astronomical unit and is abbreviated as

AU. If these units are used in Kepler's 3rd Law,

the denominators in the preceding equation are

numerically equal to unity and it may be written

in the simple form - P (years)2 R (AUs)3
- This equation may then be solved for the period P

of the planet, given the length of the semi-major

axis, - P (years) R (AU)3/2
- or for the length of the semi-major axis, given

the period of the planet, - R (AU) P (Years) 2/3

Calculations of Keplers Third Law

- Let's calculate the "radius" of the orbit of Mars

(that is, the length of the semi-major axis of

the orbit) from the orbital period. - The time for Mars to orbit the Sun is observed to

be 1.88 Earth years. - R P 2/3 (1.88) 2/3 1.52 AU
- As a second example, let us calculate the orbital

period for Pluto, given that its observed average

separation from the Sun is 39.44 astronomical

units. From Kepler's 3rd Law - P R3/2 (39.44)3/2 248 Years

How does it Relate to Satellites?

- We will now address the following questions
- Does all this apply to satellites orbiting the

Earth? - How can we use the Keplerian equations to find

the position of a satellite? - How do satellites orbit around the Earth?
- How do we send a satellite in orbit?

Orbital Mechanics

- Orbital mechanics is the study of the motions of

artificial satellites and space vehicles moving

under the influence of forces such as gravity,

atmospheric drag, thrust, etc. - Orbital mechanics is a modern offshoot of

celestial mechanics which is the study of the

motions of natural celestial bodies such as the

moon and planets. - The root of orbital mechanics can be traced back

to the 17th century when mathematician Isaac

Newton (1642-1727) put forward his laws of motion

and formulated his law of universal gravitation. - The engineering applications of orbital mechanics

include ascent trajectories, reentry and landing,

rendezvous computations, and lunar and

interplanetary trajectories.

Orbital Mechanics II

- Orbital mechanics remain a mystery to most people
- Difficulty in thinking in 3D
- Cryptic names given by astronomers
- To make matters worse, sometimes several

different names are used to specify the same

number. - Vocabulary is one of the hardest part of

celestial mechanics!

Perigee and Apogee

- The point where the secondary is closest to the

primary is called perigee, although it's

sometimes called periapsis or perifocus. - The point where the seconday is farthest from

primary is called apogee (aka apoapsis, or

apifocus).

Vernal Equinox

- For some of our calculations we will use the term

vernal equinox - Teachers have told children for years that the

vernal equinox is "the place in the sky where the

sun rises on the first day of Spring". - This is a horrible definition. Most teachers, and

students, have no idea what the first day of

spring is (except a date on a calendar), and no

idea why the sun should be in the same place in

the sky on that date every year.

Vernal Equinox II

- Consider the orbit of the Sun around the Earth.

Although the Earth does not orbit around the sun,

the math is equally valid either way, and it

suits our needs at this instant to think of the

Sun orbiting the Earth. - The orbit of the sun has an inclination of about

23.5 degrees. (Astronomers use an infinitely more

obscure name The Obliquity of The Ecliptic.) - The orbit of the Sun is divided (by humans) into

four equally sized portions called seasons. - In other words, the first day of Spring is the

day that the sun crosses through the equatorial

plane going from South to North.

Keplerian Elements

- Severn numbers are required to define a satellite

orbit. This set of seven numbers is called the

satellite orbital elements, or sometimes

"Keplerian" elements - These numbers define an ellipse, orient it about

the Earth, and place the satellite on the ellipse

at a particular time. In the Keplerian model,

satellites orbit in an ellipse of constant shape

and orientation - The real world is slightly more complex than the

Keplerian model, and tracking programs compensate

for this by introducing minor corrections to the

Keplerian model - These corrections are known as perturbations, and

are due to the unevenness of the earth's

gravitational field (which luckily you don't have

to specify), and the "drag" on the satellite due

to atmosphere. - Drag becomes an optional eighth orbital element

Keplerian Elements II

- Epoch
- Orbital Inclination
- Right Ascension of Ascending Node (R.A.A.N.)
- Argument of Perigee
- Eccentricity
- Mean Motion
- Mean Anomaly
- Drag (optional)

Epoch

- A set of orbital elements is a snapshot, at a

particular time, of the orbit of a satellite. - Epoch is simply a number which specifies the time

at which the snapshot was taken.

Orbital Inclination

- The orbit ellipse lies in a plane known as the

orbital plane. The orbital plane always goes

through the center of the earth, but may be

tilted any angle relative to the equator.

Inclination is the angle between the orbital

plane and the equatorial plane. - By convention, inclination is a number between 0

and 180 degrees. - Orbits with inclination near 0 degrees are called

equatorial orbits, or Geostationary. Orbits with

inclination near 90 degrees are called polar. - The intersection of the equatorial plane and the

orbital plane is a line which is called the line

of nodes.

Right Ascension of Ascending Node

- Two numbers orient the orbital plane in space.

The first number was Inclination, RAAN is the

second. - After we've specified inclination, there are

still an infinite number of orbital planes

possible. The line of nodes can intersect

anywhere along the equator. - If we specify where along the equator the line of

nodes intersects, we will have the orbital plane

fully specified. - The line of nodes intersects two places, of

course. We only need to specify one of them. One

is called the ascending node (where the satellite

crosses the equator going from south to north).

The other is called the descending node (where

the satellite crosses the equator going from

north to south). - By convention, we specify the location of the

ascending node.

Right Ascension of Ascending Node II

- The Earth is spinning. This means that we can't

use the common latitude/longitude coordinate

system to specify where the line of nodes points. - Instead, we use an astronomical coordinate

system, known as the right ascension /

declination coordinate system, which does not

spin with the Earth. - Right ascension is another fancy word for an

angle, in this case, an angle measured in the

equatorial plane from a reference point in the

sky where right ascension is defined to be zero. - Astronomers call this point the vernal equinox.
- Finally, "right ascension of ascending node" is

an angle, measured at the center of the earth,

from the vernal equinox to the ascending node.

Right Ascension of Ascending Node III

- Draw a line from the center of the earth to the

point where our satellite crosses the equator

(going from south to north). If this line points

directly at the vernal equinox, then RAAN 0

degrees. - By convention, RAAN is a number in the range 0 to

360 degrees.

Argument of Perigee

- Argument is yet another fancy word for angle. Now

that we've oriented the orbital plane in space,

we need to orient the orbit ellipse in the

orbital plane. We do this by specifying a single

angle known as argument of perigee. - If we draw a line from perigee to apogee, this

line is called the line-of-apsides or major-axis

of the ellipse (Green dotted line).

Argument of Perigee II

- The line-of-apsides passes through the center of

the Earth. - We've already identified another line passing

through the center of the earth the line of

nodes. - The angle between these two (green dotted) lines

is called the argument of perigee. The argument

of perigee is the angle (measured at the center

of the earth) from the ascending node to perigee. - Example When ARGP 0, the perigee occurs at the

same place as the ascending node. That means that

the satellite would be closest to earth just as

it rises up over the equator. When ARGP 180

degrees, apogee would occur at the same place as

the ascending node. That means that the satellite

would be farthest from earth just as it rises up

over the equator. - By convention, ARGP is an angle between 0 and 360

degrees.

Eccentricity

- In the Keplerian orbit model, the satellite orbit

is an ellipse. Eccentricity tells us the "shape"

of the ellipse. - For our purposes eccentricity must be in the

range 0 lt e lt 1.

Mean Motion

- So far we've found the orientation of the orbital

plane, the orientation of the orbit ellipse in

the orbital plane, and the shape of the orbit

ellipse. - Now we need to know the "size" of the orbit

ellipse. In other words, how far away is the

satellite? - Kepler's third law of orbital motion gives us a

precise relationship between the speed of the

satellite and its distance from the earth. - Satellites that are close to the earth orbit very

quickly. Satellites far away orbit slowly. This

means that we could accomplish the same thing by

specifying either the speed at which the

satellite is moving, or its distance from the

Earth! - Satellites in circular orbits travel at a

constant speed. We just specify that speed, and

we're done. Satellites in non-circular (i.e.,

eccentricity gt 0) orbits move faster when they

are closer to the Earth, and slower when they are

farther away. - The common practice is to average the speed. You

could call this number "average speed", but

astronomers call it the "Mean Motion". - Mean Motion is usually given in units of

revolutions per day.

Mean Motion II

- In this context, a revolution or period is

defined as the time from one perigee to the next. - Sometimes "orbit period" is specified as an

orbital element instead of Mean Motion. Period is

simply the reciprocal of Mean Motion. A satellite

with a Mean Motion of 2 revs per day, for

example, has a period of 12 hours. - Sometimes semi-major-axis (SMA) is specified

instead of Mean Motion. SMA is one-half the

length (measured the long way) of the orbit

ellipse, and is directly related to mean motion

by a simple equation. - Typically, satellites have Mean Motions in the

range of 1 rev/day to about 16 rev/day.

Mean Anomaly

- Now that we have the size, shape, and orientation

of the orbit firmly established, the only thing

left to do is specify where exactly the satellite

is on this orbit ellipse at some particular time.

- Our very first orbital element (Epoch) specified

a particular time, so all we need to do now is

specify where, on the ellipse, our satellite was

exactly at the Epoch time. - Anomaly is yet another astronomer-word for angle!
- Mean anomaly is simply an angle that marches

uniformly in time from 0 to 360 degrees during

one revolution. - It is defined to be 0 degrees at perigee, and

therefore is 180 degrees at apogee.

Mean Anomaly II

- If you had a satellite in a circular orbit

(therefore moving at constant speed) and you

stood in the center of the Earth and measured

this angle from perigee, you would point directly

at the satellite. - Satellites in non-circular orbits move at a

non-constant speed, so this simple relation

doesn't hold. This relation does hold for two

important points on the orbit, however, no matter

what the eccentricity. Perigee always occurs at

MA 0, and apogee always occurs at MA 180

degrees.

Drag

- Drag caused by the Earth's atmosphere causes

satellites to spiral downward. As they spiral

downward, they speed up. - The Drag orbital element simply tells us the rate

at which Mean Motion is changing due to drag or

other related effects. - Drag is one half the first time derivative of

Mean Motion. - Its units are revolutions per day per day. It is

typically a very small number. - Common values for low-Earth-orbiting satellites

are on the order of 10-4. - Common values for high-orbiting satellites are on

the order of 10-7 or smaller. - Can you tell me why?

Drag II

- Occasionally, published orbital elements for a

high-orbiting satellite will show a negative

Drag! - There are several potential reasons for negative

drag. - First, the measurement which produced the orbital

elements may have been in error. - A satellite is subject to many forces besides

Earth's gravity and atmospheric drag - Some of these forces (for example gravity of the

Sun and Moon) may act together to cause a

satellite to be pulled upward by a very slight

amount. - This can happen if the Sun and Moon are aligned

with the satellite's orbit in a particular way.

If the orbit is measured when this is happening,

a small negative Drag term may actually provide

the best possible 'fit' to the actual satellite

motion over a short period of time.

Drag III

- You typically want a set of orbital elements to

estimate the position of a satellite reasonably

well for as long as possible, often several

months. Negative Drag never accurately reflects

what's happening over a long period of time. Some

programs will accept negative values for Drag,

but I don't approve of them. Feel free to

substitute zero in place of any published

negative Drag value.

Other Satellite Parameters

- The following parameters are optional. They allow

tracking programs to provide more information

that may be useful or fun

Epoch Rev

- This tells the tracking program how many times

the satellite has orbited from the time it was

launched until the time specified by "Epoch". - Epoch Rev is used to calculate the revolution

number displayed by the tracking program. Don't

be surprised if you find that orbital element

sets which come from NASA have incorrect values

for Epoch Rev.

Attitude

- The spacecraft attitude is a measure of how the

satellite is oriented in space. Hopefully, it is

oriented so that its antennas point toward you! - There are several orientation schemes used in

satellites. The Bahn coordinates apply only to

spacecraft which are spin-stablized.

Spin-stabilized satellites maintain a constant

inertial orientation, i.e., its antennas point a

fixed direction in space. - The Bahn coordinates consist of two angles, often

called Bahn Latitude and Bahn Longitude. Ideally,

these numbers remain constant except when the

spacecraft controllers are re-orienting the

spacecraft. In practice, they drift slowly.

How do Satellite Elements look like?

- Satellite elements can be downloaded from NORAD

http//www.celestrak.com/NORAD/elements/ - They are in TLE format (Two Line Elements)
- NOAA 18
- 1 28654U 05018A 06216.35688869 -.00000204 00000-0

-89227-4 0 5825 - 2 28654 98.7889 158.6074 0015435 83.8593 276.4346

14.10977823 62171

TLE Format

- Data for each satellite consists of three lines

in the following format - AAAAAAAAAAAAAAAAAAAAAAAA1 NNNNNU NNNNNAAA

NNNNN.NNNNNNNN .NNNNNNNN NNNNN-N NNNNN-N N

NNNNN2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN

NNN.NNNN NN.NNNNNNNNNNNNNN - Line 0 is a twenty-four character name

Satellite Orbit

- One of the most important aspect of a satellite

orbit is its inclination - The inclination limits the types of coverage and

data that a satellite can acquire - The velocity of the satellites determines the

height above the geoid

Satellites Orbit

Geosyncronous Satellites

- GEO are circular orbits around the Earth having a

period of 24 hours. - A geosynchronous orbit with an inclination of

zero degrees is called a geostationary orbit. - A spacecraft in a geostationary orbit appears to

hang motionless above one position on the Earth's

equator. For this reason, they are ideal for some

types of communication and meteorological

satellites. - A spacecraft in an inclined geosynchronous orbit

will appear to follow a regular figure-8 pattern

in the sky once every orbit. - To attain geosynchronous orbit, a spacecraft is

first launched into an elliptical orbit with an

apogee of 35,786 km (22,236 miles) called a

geosynchronous transfer orbit (GTO). The orbit is

then circularized by firing the spacecraft's

engine at apogee.

Typical Geostationary Coverage

Metereological Satellites

World Clouds

Polar Orbits

- PO are orbits with an inclination of 90 degrees.
- Polar orbits are useful for satellites that carry

out mapping and/or surveillance operations

because as the planet rotates the spacecraft has

access to virtually every point on the planet's

surface - Most PO are circular to slightly elliptical at

distances ranging from 700 to 1700 km (435 - 1056

mi) from the geoid. - At different altitudes they travel at different

speeds.

(Near) Polar Orbiting Satellites

Ascending Vs. Descending

Daily Coverage

Polar Regions

- The satellite doesn't pass directly over the pole

due to the slight inclination of the orbital

plane. - The transparent overlay identifies the 3000 km

wide swath that is viewed by the AVHRR imaging

instrument on the satellite. - The yellow curves delineate the limits of the 60

degree viewing arcs from the six "standard"

geostationary satellites included in these

discussions.

Multiple Passes

Sun Synchronous Orbits

- SSO are near polar orbits where a satellite

crosses periapsis at about the same local time

every orbit. - This is useful if a satellite is carrying

instruments which depend on a certain angle of

solar illumination on the planet's surface. - In order to maintain an exact synchronous timing,

it may be necessary to conduct occasional

propulsive maneuvers to adjust the orbit. - Most research satellites are in Sun Syncronous

Orbits - There is a special kind of sun-synchronous orbit

called a dawn-to-dusk orbit. In a dawn-to-dusk

orbit, the satellite trails the Earth's shadow

(Why do you think this could be convinient?)

Molniya Orbits

- They are highly eccentric Earth orbits with

periods of approximately 12 hours (2 revolutions

per day). - The orbital inclination is chosen so the rate of

change of perigee is zero, thus both apogee and

perigee can be maintained over fixed latitudes. - This condition occurs at inclinations of 63.4

degrees and 116.6 degrees. For these orbits the

argument of perigee is typically placed in the

southern hemisphere, so the satellite remains

above the northern hemisphere near apogee for

approximately 11 hours per orbit. This

orientation can provide good ground coverage at

high northern latitudes.

Molniya Orbits

Tundra Orbits

- Tundra orbit is a class of a highly elliptic

orbit with inclination of 63.4 and orbital

period of one sidereal day (almost 24 hours). - A satellite placed in this orbit spends most of

its time over a designated area of the earth, a

phenomenon known as apogee dwell.

Different Orbital Distances

Orbital Distances

- A low Earth orbit (LEO) is an orbit around Earth

between the atmosphere and the Van Allen

radiation belt. The boundaries are not firmly

defined but are typically around 200 - 1200 km

(124 - 726 miles) above the Earth's surface - Intermediate circular orbit (ICO), also called

Medium Earth Orbit (MEO), is used by satellites

between the altitudes of Low Earth Orbit (up to

1400 km) and geostationary orbit (35,790 km) - A rather vaguely defined orbit, which usually

means anything from geosynchronous orbit up

Satellite Constellation

- A group of electronic satellites working in

concert is known as a satellite constellation. - Such a constellation can be considered to be a

number of satellites with coordinated ground

coverage, operating together under shared

control, synchronised so that they overlap well

in coverage and complement rather than interfere

with other satellites' coverage.

Satellite Formation

Demonstration

Formalization

- In the followings we will formalize all the

concepts we have discussed - When you measure what you are speaking about and

express it in numbers, you know something about

it, but when you cannot express it in numbers

your knowledge is of a meager and unsatisfactory

kind. (Lord Kelvin, British Scientist) William

Thompson, Lord Kelvin, Popular Lectures and

Addresses 1891-1894, in Bartlett's Familiar

Quotations, Fourteenth Edition, 1968, p. 723a.

Newton's Laws of Motion and Universal Gravitation

- The first law states that if no forces are

acting, a body at rest will remain at rest, and a

body in motion will remain in motion in a

straight line. Thus, if no forces are acting, the

velocity (both magnitude and direction) will

remain constant. - The second law tells us that if a force is

applied there will be a change in velocity, i.e.

an acceleration, proportional to the magnitude of

the force and in the direction in which the force

is applied. This law may be summarized by the

equation - F ma
- where F is the force, m is the mass of the

particle, and a is the acceleration. - Remember that both F and a are vector quantities

Newton's Laws of Motion and Universal Gravitation

II

- The third law states that if body 1 exerts a

force on body 2, then body 2 will exert a force

of equal strength, but opposite in direction, on

body 1. This law is commonly stated, "for every

action there is an equal and opposite reaction". - In his law of universal gravitation, Newton

states that two particles having masses m1 and m2

and separated by a distance r are attracted to

each other with equal and opposite forces

directed along the line joining the particles.

The common magnitude F of the two forces is - where G is an universal constant, called the

constant of gravitation We will use this formula

often

Newton's Laws of Motion and Universal Gravitation

III

- Let's now look at the force that the Earth exerts

on an object. If the object has a mass m, and the

Earth has mass M, and the object's distance from

the center of the Earth is r, then the force that

the Earth exerts on the object is GmM /r2 . If we

drop the object, the Earth's gravity will cause

it to accelerate toward the center of the Earth.

By Newton's second law (F ma), this

acceleration g must equal (GmM / r2 )

/ m, or - At the surface of the Earth this acceleration has

the valve 9.80665 m/s2 - Many of the upcoming computations will be

somewhat simplified if we express the product GM

as a constant, which for Earth has the value

3.986005x1014 m3/s2

Problem

- Your professor is asking to send CEOSR-1

satellite into orbit using a rocket - The weight of the rocket satellite is 500,000

kg loaded with fuel and 320,000 kg with no fuel - The rocket creates a thrust of 10,000,000N
- Approximating g at 10m/s2, what is the

acceleration at (1) launch and at (2) burn out?

Solution

- Lets assume upward direction to be positive and

downward to be negative, so we can work with

numbers rather than vectors - At launch, two forces act on the rocket
- T Positive thrust 10,000,000N
- W Negative mg 500,000kg 10m/s2

-5,000,000N - The total Force is TW 5,000,000N
- By Newtons second law
- aF/m 5,000,000N / 500,000kg 10m/s2 10g

Solution II

- At burnout
- W Negative mg 320,000kg 10m/s2

-3,200,000N - The total Force is TW 6,800,000N
- By Newtons second law
- aF/m 6,800,000N / 320,000kg 10m/s2 21.25g

Uniform Circular Motion

- In the simple case of free fall, a particle

accelerates toward the center of the Earth while

moving in a straight line. The velocity of the

particle changes in magnitude, but not in

direction. - In the case of uniform circular motion a particle

moves in a circle with constant speed. The

velocity of the particle changes continuously in

direction, but not in magnitude. - From Newton's laws we see that since the

direction of the velocity is changing, there is

an acceleration. - This acceleration, called centripetal

acceleration is directed inward toward the center

of the circle and is given by - where v is the speed of the particle and r is the

radius of the circle. - Every accelerating particle must have a force

acting on it, defined by Newton's second law (F

ma). Thus, a particle undergoing uniform circular

motion is under the influence of a force, called

centripetal force, whose magnitude is given by - The direction of F at any instant must be in the

direction of a at the same instant, that is

radially inward.

Uniform Circular Motion II

- A satellite in orbit is acted on only by the

forces of gravity. - The inward acceleration which causes the

satellite to move in a circular orbit is the

gravitational acceleration caused by the body

around which the satellite orbits. - Hence, the satellite's centripetal acceleration

is g, that is g v2/r. - From Newton's law of universal gravitation we

know that g GM /r2. Therefore, by setting these

equations equal to one another we find that, for

a circular orbit,

Problem

- Calculate the velocity of a satellite orbiting

the Earth in a circular orbit at an altitude of

200 km above the Earth's surface. - Radius of Earth 6,378.140 km
- GM of Earth 3.986005x1014 m3/s2
- Given r (6,378.14 200) x 1,000 6,578,140 m
- v SQRT GM / r
- v SQRT 3.986005x1014 / 6,578,140
- v 7,784 m/s

Motions of Planets and Satellites

- Now we are going to formalize Newtons three laws
- 1. All planets move in elliptical orbits with

the sun at one focus. 2. A line joining any

planet to the sun sweeps out equal areas in equal

times. 3. The square of the period of any

planet about the sun is proportional to the cube

of the planet's mean distance from the sun.

Motions of Planets and Satellites II

- Although all planets move in elliptical orbits,

their eccentricity is very small. We can learn

much about planetary motion by considering the

special case of circular orbits. - We shall neglect the forces between planets,

considering only a planet's interaction with the

sun. - These considerations apply equally well to the

motion of a satellite about a planet.

Motions of Planets and Satellites III

- Let's examine the case of two bodies of masses M

and m moving in circular orbits under the

influence of each other's gravitational

attraction - The center of mass lies along the line joining

them at a point C such that mr MR - M and m move in an orbit of radius R and r,

respectively, with the same angular velocity - For this to happen, the gravitational force

acting on each body must provide the necessary

centripetal acceleration. - Both forces are equal but opposite in direction.

Motions of Planets and Satellites IV

- That is, mw2r must equal Mw2R. The specific
- requirement, then, is that the gravitational

force - acting on either body must equal the centripetal
- force needed to keep it moving in its circular

orbit, - that is
- If one body has a much greater mass than the

other, as is the case of the Sun and a planet or

the Earth and a satellite, its distance from the

center of mass is much smaller than that of the

other body. - If we assume that m is negligible compared to M,

then R is negligible compared to r, then we can

simply the above formula into

Motions of Planets and Satellites V

- If we express the angular velocity in terms of

the period of revolution, w 2 /P, we obtain - where P is the period of revolution.
- This is a basic equation of planetary and

satellite motion. - It also holds for elliptical orbits if we define

r to be the semi-major axis of the orbit. - A significant consequence of this equation is

that it predicts Kepler's third law of planetary

motion, that is P2r3

Problem

- Geostationary are a special class of satellites

that orbit the Earth with a period of one day. - Anwer the following
- How will the satellite's motion appear when

viewed from the surface of the Earth? - What type of satellites use this orbit and why is

it important for them to be located in this

orbit? (Keep in mind that this is a relatively

high orbit. Satellites not occupying this band

are normally kept in much lower orbits.) - Determine the orbital radius at which the period

of a satellite's orbit will equal one day. State

your answer in - kilometers
- multiples of the Earth's radius
- fractions of the moon's orbital radius

(3) Solution

- The period of the Earth's rotation is

approximately equal to the mean solar day

(24 x 3600 s 86,400 s), but for best results

use the sidereal day (86,164 s). - We know that P2 4 x p 2 x r3 / GM
- r P2 x GM / (4 x p 2 ) 1/3
- r 86,164.12 x 3.986005x1014 / (4 x p 2 ) 1/3

- r 42,164,170 m

(3) Solution II

- Convert to Earth radii r 4.216 107 m /

6,378,140 m 6.610 7 Earth radii - Convert to Earth-Moon distances

r 4.216 107 m / 384,400,000 m9

0.1097 1 distance from earth to moon - Homework
- Answer 1 and 2
- Can you find the same solution using Keplers

third law?

What About for Elliptical Orbits?

- The figure shows a particle revolving around C

along some arbitrary path - The area swept out by the radius vector in a

short time interval t is shown shaded - This area (neglecting the small triangular region

at the end) is one-half the base times the height

or approximately r(rwDt)/2 - This expression becomes more exact as t

approaches zero, i.e. the small triangle goes to

zero more rapidly than the large one. - For any given body moving under the influence of

a central force, the value wr2 is constant

it gets a bit more complicated

- Let's now consider two points P1 and P2 in an

orbit with radii r1 and r2, and velocities v1 and

v2. - Since the velocity is always tangent to the path,

it can be seen that if f is the angle between r

and v, then - And multiplying by R
- or, for two points P1 and P2 on the orbital path
- What happens at periapsis and apoapsis, (Hint f

90 degrees)?

Consarvation of Energy

- Let's now look at the energy of the above

particle at points P1 and P2. - Conservation of energy states that the sum of the

kinetic energy and the potential energy of a

particle remains constant. - The kinetic energy T of a particle is given by

mv2/2 - The potential energy of gravity V is calculated

by the equation -GMm/r. Applying conservation of

energy we have

and finally

The eccentricity will be given by

Problem

- An artificial Earth satellite is in an elliptical

orbit which brings it to an altitude of 250 km at

perigee and out to an altitude of 500 km at

apogee. - Calculate the velocity of the satellite at both

perigee and apogee.

Solution

- Rp (6,378.14 250) x 1,000 6,628,140 m
- Ra (6,378.14 500) x 1,000 6,878,140 m
- Vp SQRT 2 x GM x Ra / (Rp x (Ra Rp))
- Vp SQRT 2 x 3.986005x1014 x 6,878,140 /

(6,628,140 x (6,878,140 6,628,140)) - Vp 7,826 m/s
- Va SQRT 2 x GM x Rp / (Ra x (Ra Rp))
- Va SQRT 2 x 3.986005x1014 x 6,628,140 /

(6,878,140 x (6,878,140 6,628,140)) - Va 7,542 m/s

Problem

- A satellite in Earth orbit passes through its

perigee point at an altitude of 200 km above the

Earth's surface and at a velocity of 7,850 m/s. - Calculate the apogee altitude of the satellite.

Solution

- Rp (6,378.14 200) x 1,000 6,578,140 m
- Vp 7,850 m/s
- Ra Rp / 2 x GM / (Rp x Vp2) - 1
- Ra 6,578,140 / 2 x 3.986005x1014 / (6,578,140

x 7,8502) - 1 - Ra 6,805,140 m
- Altitude _at_ apogee 6,805,140 / 1,000 - 6,378.14

427.0 km

Additional Problems

- What we have not taken into account
- Third-Body Perturbations
- Perturbations due to Non-spherical Earth
- Perturbations from Atmospheric Drag
- Perturbations from Solar Radiation

References

- http//library.thinkquest.org/29033/begin/orbits.h

tm - http//www.mmto.org/obscats/tle.html
- http//hypertextbook.com/physics/mechanics/orbital

-mechanics-1/ - http//www.nsbri.org/HumanPhysSpace/introduction/i

ntro-environment-gravity.html - http//www.astro-tom.com/technical_data/elliptical

_orbits.htm - http//www.braeunig.us/space/orbmech.htm
- http//www.amsat.org/amsat-new/information/faqs/ke

pmodel.php - http//www.ulo.ucl.ac.uk/students/1b30/lectures/
- http//www.rap.ucar.edu/djohnson/satellite/covera

ge.html

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