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Generic Conical Orbits, Kepler

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Title: Generic Conical Orbits, Kepler


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

Line 1 Line 1
01 Line Number of Element Data
03-07 Satellite Number
08 Classification (UUnclassified)
10-11 International Designator, last two digits of launch year,  2000 if lt 57.
12-14 International Designator, launch number of the year
15-17 International Designator, piece of the launch
19-20 Epoch Year, last two digits of year,  2000 if lt 57
21-32 Epoch Day of the year and fractional portion of the day
34-43 First Time Derivative of the Mean Motion
45-52 Second Time Derivative of Mean Motion (decimal point assumed)
54-61 BSTAR drag term (decimal point assumed)
63 Ephemeris type
65-68 Element number
69 Checksum (Modulo 10) (Letters, blanks, periods, plus signs 0 minus signs 1)
Line 2 Line 2
01 Line Number of Element Data
03-07 Satellite Number
09-16 Inclination Degrees
18-25 Right Ascension of the Ascending Node Degrees
27-33 Eccentricity (decimal point assumed)
35-42 Argument of Perigee Degrees
44-51 Mean Anomaly Degrees
53-63 Mean Motion Revs per day
64-68 Revolution number at epoch Revs
69 Checksum (Modulo 10)
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
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