eele 6335 telecom. chapter 2: system orbits and...

Dr.Mohammed Taha El Astal, IUG, EE dept, 2016

9/20/2016

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EELE 6335

Telecom. System

Part I:

Satellite Communic

ations

Winter 2016

Prepared by

Dr. Mohammed Taha El Astal

Chapter 2:

Orbits and Launching Methods

Content

Kepler’s First, Second, and Third Law

Definitions of Terms for Earth-Orbiting Satellites

Orbital Elements

Apogee and Perigee Heights

Orbit Perturbations

Inclined Orbits


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2.1 Introduction

• Satellite/Spacecraft orbiting the earth follow the same laws that govern the motion of the planets around to sun.

• Johannes Kepler (1571-1630) was able to derive empirically three laws describing planetary motion.

• Later, Isaac Newton derived Kepler's laws from his own laws of mechanics and developed the theory of gravitation.

• Kepler’s laws can be applied almost for any two bodies in space interact through gravitation.

CONT.

• interact through gravitation?

• Variables : mass1

mass 2

Distance r

(More massive) : Primary

(Less massive) : Secondary or Satellite


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• The center of mass of the two-body system (barycenter) is always centered on one of the foci.

In our case (sat.+Earth), the barycenter coincides with the center of the earth

The earth is always at one of the foci.

• The eccentricity e if given by 𝑒 =𝑎2−𝑏2

𝑎

Elliptical orbit: 0 < 𝑒 < 1.

e= 0, the orbit becomes circular.

• Refer to App. ‘B’ for details of the e and ellipse.

• The e & a are two of the orbital parameters.

2.2 Kepler’s First Law

The path followed by a satellite around the primary will be an ellipse

CONT.

𝑎𝑟𝑒𝑎 = 𝜋 × 𝑎 × 𝑏

𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 ≅ 2𝜋𝑎2 + 𝑏2

2

the tangent line has equal angles with the two lines going to each focus

𝑥

𝑎

2

+𝑦

𝑏

2

= 1

f : linear eccentricity e is the ratio of distance between two focus to the length of major axis=2f/2a=f/a

gets a more elongated

Question:

From the original

definition of e, derive the

most common

formula of e

:𝑎2−𝑏2

𝑎


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CONT.

because of the equal area law, it follows that the velocity at S2 is less than that at S1.

Mean that the satellite takes longer to travel a given distance when it is farther away from earth.

They used this property to increase the length of time a satellite can be seen from particular geographic regions of the earth.

2.3 Kepler’s Second Law

For equal time intervals, a satellite will sweep out equal areas in its orbital plane, focused at the barycenter


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• Mean distance the arithmetic mean of the greatest and least distances of a satellite from the earth.

• Mean distance = semimojor axis (a)

2.4 Kepler’s Third Law

The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies: 𝑃2/𝑎3 is constant value

• Orbital period: 𝑃 =2𝜋

𝑛 (n in rad/sec),

where, n is the mean motion of the satellite in rad/sec

• Mean motion: it is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body.

N.B. This equation applies only to the ideal situation : a perfectly spherical earth of uniform mass, with no perturbing forces acting, such as atmospheric drag. (see Sec. 2.8)

Third Law become as: 𝑎3 =𝜇

𝑛2,

where, 𝜇 is the earth’s geocentric gravitational constant = 3.986005 × 1014 𝑚3/𝑠2

CONT.(3rd law: another perspective)

In a way, you can deduce directly that 𝑃2/𝑎3 is constant value for any planetary (satellite/spacecraft) motion, this mean?

• a++P++

large ellipse/orbit results longer time to complete a period (in other words, slower motion)

• a--P--

Smaller ellipse/orbit results shorter time to complete a period (in other words, faster motion)


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• Subsatellite path: this is the path traced out on the earth’s surface directly below the satellite.

• Apogee: the point farthest from earth. Apogee height (ha)

• Perigee: The point of closest approach to earth. Perigee height (hp).

• Line of apsides: The line joining the perigee and apogee through the center of the earth.

2.5 Definition of Terms of Earth-Orbiting Satellites:

CONT.

• Ascending node: The point where the orbit crosses the equatorial plane going from south to north.

• Descending node: The point where the orbit crosses the equatorial plane going from north to south.

• Line of nodes: The line joining the ascending and descending nodes through the center of the earth.


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CONT.

• Inclination (i): the angle between the orbital plane and the earth’s equatorial plane.

• It is measured at the ascending node from the equator to the orbit, going from east to north.

• i relate to subsatellite path?? • It will be seen that the greatest

latitude, north or south, reached by the subsatellite path is equal to the inclination.

CONT.

• Prograde orbit: an orbit in which the satellite moves in the same direction as the earth’s rotation. (also known as a direct orbit) • i of a prograde orbit lies between 0° - 90°.

• Most satellites are launched in a prograde orbit , why??

• because the earth’s rotational velocity provides part of the orbital velocity with a consequent saving in launch energy.

• Retrograde orbit: An orbit in which the satellite moves in a direction counter to the earth’s rotation.

• i always lies between 90° and 180°.


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CONT.

• Up : mean prograde orbit

• Down : mean retrograde

• Exercise : do it for Earth, Uranus, and Venus

CONT.

• Argument of perigee (𝝎): the angle from ascending node to perigee, measured in the orbital plane at the earth’s center, in the direction of satellite motion.

• Mean anomaly (M): gives an average value of the angular position of the satellite with reference to the perigee.

For a circular orbit, M gives the angular position For elliptical orbit, the position is much more

difficult to calculate, and M is used just as an intermediate step.

True anomaly: is the angle from perigee to the satellite position, measured at the earth’s center.

This gives the true angular position of the satellite in the orbit as a function of time.


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• Earth-orbiting artificial satellites are defined by six orbital elements referred to as the keplerian element set.

2.6 Orbital Elements:

(a & e ) give the shape of the ellipse

(v or M) gives the position of the satellite in its orbit at a reference time known as the epoch

(I & Ω) relate the orbital plane’s position to the earth.

𝜔 gives the rotation of the orbit’s perigee point relative to the orbit’s line of nodes in the earth’s equatorial plane.


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CONT.

• Appendix C lists the two-line elements provided to users by NASA (see Celestrak site).

• Using suitable prediction formula, the state (position and velocity) at any point in the past or future can be estimated to some accuracy.

• Because of :

1. the equatorial bulge causes slow variations in w and Ω

2. and other perturbing forces may alter the orbital elements slightly,

the values are specified for the reference time or epoch (called as two-line elements (TLE))

• A two-line element set (TLE) is a data format encoding a list of orbital elements of an Earth-orbiting object for a given point in time (the epoch).

CONT.

• An example TLE for the International Space Station:

• The meaning of this data is as follows:(see attached cases study file)

• Ref: https://en.wikipedia.org/wiki/Two-line_element_set

• Note: the semimajor axis is not specified, but this can be calculated from the data given.

https://en.wikipedia.org/wiki/Two-line_element_set















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2.7 Apogee and Perigee Heights

• Although not specified as orbital elements, they are often required.

• As shown earlier : 𝑟𝑎 = 𝑎 1 + 𝑒 𝑟𝑝 = 𝑎 1 − 𝑒

• To find ℎ𝑎 and ℎ𝑝 , the radius of the earth must be subtracted from the radii lengths


Next time

2.8 Orbit Perturbations 2.9 Inclined Orbits






















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Dr. Mohammed Taha El Astal [email protected] [email protected]

20/9/2016

eele 6335 telecom. chapter 2: system orbits and...

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