Generic Conical Orbits, Keplers Laws, Satellite Orbits and Orbital Mechanics

1
Generic Conical Orbits, Keplers Laws,Satellite
Orbits and Orbital Mechanics

• Guido Cervone

2
Summary

• Class Website
• Continuation of previous lecture
• Generic Conical Orbits, Keplers Laws,Satellite
  Orbits and Orbital Mechanics
• Video - Satellite Launch

3
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

4
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

5
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

6
How do Satellites Orbit?
7
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.

8
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.

9
Conic Orbits II
10
Conic Orbits III

• Most satellite and planetary orbits are elliptical

11
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.

12
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.

13
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

14
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.

15
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.

16
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.

17
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

18
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

19
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?

20
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.

21
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!

22
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).

23
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.

24
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.

25
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

26
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)

27
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.

28
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.

29
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.

30
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.

31
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.

32
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).

33
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.

34
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.

35
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.

36
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.

37
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.

38
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.

39
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?

40
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.

41
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.

42
Other Satellite Parameters

• The following parameters are optional. They allow
  tracking programs to provide more information
  that may be useful or fun

43
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.

44
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.

45
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

46
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

47
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

48
Satellites Orbit
49
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.

50
Typical Geostationary Coverage
51
Metereological Satellites
52
World Clouds
53
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.

54
(Near) Polar Orbiting Satellites
55
Ascending Vs. Descending
56
Daily Coverage
57
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.

58
Multiple Passes
59
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?)

60
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.

61
Molniya Orbits
62
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.

63
Different Orbital Distances
64
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

65
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.

66
Satellite Formation
67
Demonstration
68
Formalization

• In the following slides 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.

69
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

70
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

71
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

72
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?

73
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

74
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

75
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.

76
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,

77
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

78
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.

79
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.

80
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.

81
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

82
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

83
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

84
(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

85
(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?

86
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

87
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)?

88
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

89
and finally
The eccentricity will be given by
90
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.

91
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

92
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.

93
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

94
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

95
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