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Satellite observation systems and reference systems (ae4-e01)

Orbit Mechanics 1

E. Schrama

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Orbital Mechanics

• Reference material is Seeber, chapter 3, study only the material covered in this lecture.

• Basic Astronomy (this is where orbital mechanics originates from, so it is a good starting point, it is also directly related to reference systems).

• Orbital mechanics and Kepler orbital elements: briefly summarizes the main results of the BSc aerospace space lectures.

• Ground tracks of satellites and Visibility of satellites: this is what you must understand for satellite observation from Earth.

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Basic Astronomy• Planetary sciences

– Structure inner solar system

– Structure outer solar system

– Four big giants

• Terminology:– Difference between stars, planets, comets and galaxies

• Relation to reference systems• Relation to other lectures:

– Physics of the Earth (ae4-876):

– Planetary sciences (ae4-890)

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Inner Solar System

• Sun• Mercury• Venus• Earth + 1 Moon• Mars + 2 Moons

(Source: www.fourmilab.ch/cgi-bin/uncgi/Solar)

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Full Solar System

• Jupiter + 16 moons + ...• Saturn + 18 moons + ...• Uranus + 21 Moons + ...• Neptune + 8 Moons + ...• Pluto + 1 moon (dwarf planet)• Comets• Meteorites• Asteroid belt• Kuiper Belt• Oort Cloud• Interplanetary medium

http://seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets.html

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Four big giants: Neptune, Uranus, Saturn, Jupiter

Courtesy: JPL

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Difference planet and star

Earth

Planet

Star

Definition parsec

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Difference planet and star

• For an Earth bound observer stars follow daily circular paths relative to the pole star

• Planets are wandering between stars

• Comets are faint spots wandering between stars.

• Fixed faint spots are galaxies, there is a Messier catalog

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Path of Mars in Opposition

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Comets

Kohoutek (1974)

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Nearest galaxy: Andromeda Spiral

The largest galaxy in our group is called the Andromeda Spiral. A large spiral similar to the Milky Way. It is about 2.3 million light years from Earth and contains about 400 billion stars.

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Our galaxy

250 billion stars

100 000 ly diameter

center 30000 ly

thickness 700 ly

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Web links

Click on any of the following links:

• Powers of ten

• University of Arizona Lunar and Planetary Laboratory

• Fourmilab Switzerland

• JPL web site

• ESA web site

• CNES web site

The above links are just examples, there are many more astronomy web sites.

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Relation to reference systems

• There is a direct relation to motions of the stars and reference systems

• With the knowledge acquired in previous lectures you should be able to draw paths of stars in the sky at night

• And you should be able to explain the consequences of precession, nutation, polar motion, right ascension and declination.

• In addition you should be able to draw the local meridian, the zero meridian, the zenith, your local latitude and longitude and the horizon.

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Orbital Mechanics• Different views on the solar system by:

– Nicolaus Copernicus, – Tycho Brahe– Johannes Kepler

• Final conclusion was: – orbits of planets are just like any other object accelerating in a

gravitational field (gun bullet physics)

• Kepler’s laws on orbit motions summarize the findings for small particles in a central force field

• It is a good starting point in discussions of satellite observation systems

• In the second lecture will become more complicated

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Copernicus, Brahe and Kepler

• In the 16th century, the Polish astronomer Nicolaus Copernicus replaced the traditional Earth-centered view of planetary motion with one in which the Sun is at the center and the planets move around it in circles.

• Although the Copernican model came quite close to correctly predicting planetary motion, discrepancies existed. This became particularly evident in the case of the planet Mars, whose orbit was very accurately measured by the Danish astronomer Tycho Brahe

• The problem was solved by the German mathematician Johannes Kepler, who found that planetary orbits are not circles, but ellipses.

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Physics of bullets

dragFgy It doesn’t seem to look like Kepler’s solution

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Physics of satellite orbits

g

v

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Kepler’s Laws

• Law I: Each planet revolves around the Sun in an elliptical path, with the Sun occupying one of the foci of the ellipse.

• Law II: The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time.

• Law III: The squares of the planets' orbital periods are proportional to the cubes of the semimajor axes of their

orbits. • Reference: http://observe.ivv.nasa.gov/nasa/education/reference/orbits/orbit_sim.html

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Kepler’s first law

)cos(1

)1()(

2

e

ear

r

There are 4 cases:

• e=0, circle

• 0<e<1, ellipse

• e=1, parabola

• e>1, hyperbola

Focal point ellipse

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In-plane Kepler parameters

PeriheliumApohelium

)1( ear )1( ear

ae

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Kepler’s second law

D

B

O

C

A

ABO CDO

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Kepler’s Third law

nT

2GMan 32

The variable n represents the mean motion in radians per time unit, a is the semi-major axis, G is the gravitational constant, M is the mass of the Sun, T is the orbital period of the satellite

You should be able to scale this equation in different length and mass units.

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Keplerian orbit elements

• Equations of motions for the Kepler problem• Why is there an orbital plane?• Position and velocity in the orbital plane• Orientation of orbital plane in 3D inertial space• Kepler equation• Kepler elements, computational scheme• Once again: reference systems.

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Mechanics of Kepler orbits (1)

TzyxX

zyxr

rr

GMU

UX

),,(

222

z

y

x

r

GM

z

y

x

3

Note: x,y and z are inertial coordinates

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Mechanics and Kepler orbits (2)

plane. orbitalan todconstraine

ismotion theand in time changenot can result a As

0

.

.

.

:givesmotion of equations theofon substituti and timeto

ation Differenti vector.momentumangular theis where

thatknow wemechanics From

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3

3

H

H

z

y

x

z

y

x

r

GM

rGMz

rGMy

rGMx

z

y

x

XX

XXXXt

XX

t

H

H

XXH

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Mechanics and Kepler orbits (3)

• A particle moves in a central force field

• The motion takes place within an orbital plane

• The solution of the equation of motion is represented in the orbital plane

• Substitution 1: polar coordinates in the orbital plane

• Substitution 2: replace r by 1/u

• Characteristic solution of a mathematical pendulum

• See also Seeber page 54 t/m 66

• Eventually: transformation orbital plane to 3D

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Xi

Yi

ZiPerigee

Right ascensionNodal line

I

See Seeber p 69

Orientation ellipse in inertial coordinate system

vrH r

v Satellite XYZ: inertial cs

: right ascension

: argument van perigee

: true anomaly

I: Inclination orbit plane

H: angular momentum vector

r: position vector satellite

v: velocity satellite

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Velocity and Position

arGMv

Eeae

ear

12

)cos1(cos1

)1( 2

radius r

velocity v

Note: in this case only , or E or M depend on time.

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Kepler’s equation• There is a difference between the definition of the true anomaly, the

eccentric anomaly E and the mean anomaly M• Note: do not confuse E and the eccentricity parameter e

E

See also Seeber pg 62 ev:

M = E - e sin(E)

M = n (t - t0)

This is Kepler’s equation

PeriheliumeE

Ee

cos

sin1tan

2

Second relation:

Virtual circle

ellipse

Focus

Center

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Keplerian elements

• Position and velocity are fully described by:– The semi major axis a

– The eccentricity e

– The inclination of the orbital plane I

– The right ascension of the ascending node

– The argument van perigee – An anomalistic angle in the orbit plane (mean anomaly

M, Eccentric anomaly E or true anomaly )

• (Memorize a drawing)

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Plate 3.10 from Seeber

Note: v in Seeber is here

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Computational scheme ),,,,,(),,,,,( Ieazyxzyx

1. It is almost ALWAYS possible to solve this problem

2. The method is explained in Seeber p 96-101

• Make use of the in-plane solution

• Use rotation matrices to orient orbital plane

• Solve Kepler’s equation to relate time to theta

3. There may be singularities in this problem and you should be smart enough to find them yourself

4. There are so-called non-singular solutions to this problem

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Xe

Ye

ZePerigee

Right ascensionNodal line

I

Zie Seeber p 69

Orientation in an Earth fixed coordinate system

vrH r

v Satellite (XYZ)e: Earth fixed cs

: right ascension

: G.A.S.T.

: argument of perigee

: true anomaly

I: Inclination

H: angular moment vector

r: position vector

v: velocity satellite

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GroundtracksA ground track is a projection of a satellite on the Earth’s surface, usually we get to see sinus like patterns that slowly propagate to the West because of Earthrotation.

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VisibilityIn general: a satellite is visible when it is above the local horizon

N

E

S

WN

E

Z

Topocentric Geografic