Spacecraft orbits and missions

(1)

General Astrophysics and Space Research Course 210142, Space Physics Module Spring 2009, Joachim Vogt

Spacecraft orbits and missions

Topics of this lecture

Basics of celestial mechanics

Geocentric orbits from LEO to GEO

SSO, Lagrange points, gravity assists

Appendix

Review questions and further reading

Additional problems and sample solutions

1

Basics of celestial mechanics

Celestial mechanics:

motions and gravitational eﬀects of celestial objects (stars, planets,

moons, . . . ). The motion of planets around the Sun is described by Kepler’s laws.

Orbital mechanics or astrodynamics:

motions of rockets, man-made satellites and spacecraft. Special

atten-tion is given to orbits around the Earth, but the ﬁeld also deals with propulsion, transition between orbits, and interplanetary missions. Satellites:

objects in orbit around a celestial body (star, planet, moon). Distinguish

between

natural satellites: planets around the Sun, moons around planets,

moonlets around asteroids, and

artiﬁcial satellites: spacecrafts in orbit around a celestial body.

Orbital motion is also called revolution that has to be distinguished from planetary rotation:

the Earth revolves around the Sun, and rotates around an axis that

intersects the surface at the geographic poles.

2

Orbits and missions: Basics of celestial mechanics

Kepler I : The Law of Orbits

[Hyperphysics (1)]

Planetary orbits are elliptical, with the Sun at one focus. Important terms:

foci of the ellipse, semi-major axis, semi-minor axis, eccentricity, perihelion, aphelion.

The special case of zero eccentricity yields a circular orbit.

Kepler II : The Law of Areas

[Hyperphysics (1)]

An (imaginary) line connecting the planet to the Sun sweeps out equal areas in equal times.

The second law follows from the conser-vation of angular momentum

L = mr × v

in central force ﬁelds such as the gravi-tational ﬁeld.

Planetary motion is

fastest at the perihelion, and slowest at the aphelion.

(2)

Kepler III : The Law of Periods

[Hyperphysics (1)]

For a planet revolving around the Sun, the square of its orbital period T is proportional to the

cube of the semi-major axisa of

its elliptical orbit:T2_{∝ a}3_.

More precisely, we write

T2 ₌ 4π2

GMa

3 _.

Here GM is the product of the gravitational constant G and the

massM of the Sun.

The planetary parameters T and a allow to deduce the solar massM.

5

Exercise: The mass of the Sun

Sample question: Using the Earth’s orbital parameters, compute

the mass of the Sun. Note that the orbit is almost circular

(eccen-tricity

e = 0.0167), so you may write a = 1 AU = 1.496 · 10

8

km.

The Earth’s orbital period is T/s = 365.25 · 24 · 3600 = 3.156 · 107_{, and the}

numerical value of the gravitational constant isG = 6.673 · 10−11_m3_kg−1_s−2_.

Hence we get for the mass of the SunM = 4π2_{· a}3_/(GT2_{) in SI units, i.e.,}

M

kg =

4· 9.87 · (1.496 · 1011₎3

6.673 · 10−11· (3.156 · 107₎2 = 1.99 · 10 30_.

Question A: Repeat this exercise with the orbital parameters

of other planets to verify the result.

6

Orbits and missions: Basics of celestial mechanics General two-body problem

Kepler’s laws apply also to objects that orbit celestial bodies other than the Sun (e.g., satellites around planets) as long as the central body is much more massive than the satellite.

General case: two bodies of masses M1 and M2 orbit around the common barycenter (center-of-mass), and the distance parameters are measured with respect to it. The general form of the third law is given by

T2 ₌ 4π2

G(M1+M2)a 3 _.

Further terminology

Other terms for the points of closest approach (farthest excursion) on an elliptical orbit are as follows.

General: periaspsis or pericenter (apoapsis or apocenter). Earth: perigee (apogee).

Star: periastron (apoastron).

Moon: periselene or perilune (aposelene or apolune).

Special terms also for other planets, galaxy, black holes, . . .

Exercise: Barycenter of the Earth-Moon system

Sample question: Compute the distance

D

_bc

of the barycenter of

the Earth-Moon system from the Earth’s center-of-mass. Note

that for two masses

_M

₁

and

M

₂

located on a line at distances

D

1

and

D

2

,

D

bc

= (

M

1

D

1

+

M

2

D

2

)

/(M

1

+

M

2

).

Let the subscripts 1 and 2 belong to Earth and Moon parameters, respectively. If we measure distances from the Earth’s center, thenD1= 0, and the formula

can be divided by M2 to yield Dbc = D2/(μ + 1) where μ = M1/M2 is the

Earth-Moon mass ratio. With μ = 81, and D2 = 3.84 · 105km, we obtain

Dbc = 4680 km. The barycenter of the Earth-Moon system is inside the Earth

about 1700 km below the surface.

Question B: Repeat this exercise for other planet-moon pairs

in the solar system.

(3)

Orbital elements

[Wikipedia (2)]

To completely characterize a Keplerian (unperturbed) orbit, six parameters must be speciﬁed.

Keplerian elements:

semi-major axis a, eccentricity e, inclination i,

argument of the periapsis ω, longitude of ascending node Ω, mean anomaly M0(at epoch).

Instead of M0, other parameters are in

use also.

9

Geocentric orbits from LEO to GEO

Important categories of satellite orbits around Earth

LEO – Low Earth Orbits

MEO – Medium Earth Orbits

GSO/GEO – Geosynchronous/Geostationary Earth Orbits

Distinguish also between

polar orbits and equatorial orbits,

circular orbits (small eccentricity) and highly elliptical orbits

(large eccentricity).

10

Orbits and missions: Geocentric orbits from LEO to GEO

Low Earth Orbit (LEO)

[NASA (3)]

Altitude range:

altitudes below about 2000 km, practical upper limit of about

1000 km due to the increased radiation exposure (Van Allen belts),

practical lower limit of about

160 km due to atmospheric drag (signiﬁcant at altitudes below 500 km).

LEO satellites:

space stations (ISS), astronomy (Hubble ST), weather monitoring, communication (Iridium), reconnaissance (spy) missions

(Keyhole, SAR-Lupe).

Exercise: Orbital periods of LEO satellites

Sample question: How large is the orbital period

T of an Earth

satellite on a circular orbit at an altitude of

h = 500 km ?

We apply Kepler’s third law T2 ₌ 4π2

GMa3 (SI units) with central body mass

M = ME = 5.98 · 1024_{kg and semi-major axis} _{a = RE + h = 6.87 · 10}6_{m to}

get T s = 2π √ 6.673 · 10−11· 5.98 · 1024(6.87 · 10 6₎3/2 _{= 5664}_.

The orbital period of the satellite is about 94 minutes.

Question C: Repeat this exercise for an Earth satellite on a

circular orbit at altitude

h = 800 km.

(4)

Equatorial and polar LEOs

Equatorial LEO: low inclination, least energy requirements.

Polar LEO: high inclination, used for Earth monitoring and

surveillance.

[GFZ/CHAMP (4)]

13

Geosynchronous Orbit and Geostationary Orbit

Geosynchronous Orbit (GSO)

Orbital periods are exactly one sidereal Earth day: T = 23h₅₆m_.

Ground paths are repeated once per day.

At the farthest point of a GSO, the distance from the Earth’s center

(semi-major axis) is_{a = 6.6 RE.}

GSO satellites are often used for telecommunication purposes.

Geostationary Orbit (GEO) or Clarke Orbit:

= circular equatorial GSO, i.e., a GSO with zero eccentricity and zero inclination.

From the ground, a satellite at a GEO always occupies the same point

in the sky.

Supersynchronous orbits: above GSO/GEO, westward drift,

are used for disposal (graveyard orbits) or storage of satellites.

Subsynchronous orbits: below GSO/GEO, eastward drift,

can be used for station changes in an eastern direction.

14

Special GSOs: Tundra orbits

Tundra orbits are examples of GSOs with high values of

inclina-tion and eccentricity.

[Chris Peat (5)]

Geostationary Transfer Orbits (GTO)

[Wikipedia (6)]

Direct insertion into GEO only by Heavy Lift Launch Vehicles (e.g., Delta IV, Space Shuttle, Proton, Ariane 5).

Launch vehicles of smaller capacity use a geostationary transfer orbit (GTO):

launch into a (circular) LEO,

upper stage of the launch vehicle ﬁres

a rocket (tangent to the orbit),

increase of velocity (delta-V) lifts the

apogee,

once in GTO, the satellite itself ﬁres

at the agogee (apogee motor) to lift the perigee.

Such a procedure is also called a Hohmann transfer.

(5)

Medium Earth Orbit (MEO)

[Wikipedia (6)]

The term Medium Earth Orbit (MEO) refers to the region in space between LEO and GEO.

The most common use for satellites in MEO is navigation (e.g., GPS, Glonass, Galileo).

Typical example: GPS

circular orbits,

orbital radius 26 600 km (altitude

20 200 km),

each satellite completes two orbits

each day.

17

Special MEOs: Molniya Orbits

A Semi-Synchronous Orbit has an orbital period of half a sidereal day (i.e.,

T = 11h₅₈m_{). Examples are the GPS constellation, and also the Russian} Molniya satellites.

[Wikipedia (6)]

18

SSO, Lagrange points, gravity assist

Kepler’s laws describe the motion of an object in the gravitational force ﬁeld of a single idealized (spherically symmetric) celestial body. Not considered in the case of geocentric satellites are, e.g.,

(a) atmospheric drag,

(b) non-spherically symmetric components of the Earth’s gravitational ﬁeld, (c) gravitational attraction of other celestial bodies (Sun, Moon, . . . ). Such perturbations give rise to variations of the orbital elements which are monitored and corrected in the process of station-keeping.

A Sun-Synchronous Orbit (SSO) is a special polar LEO that takes advan-tage of an orbit perturbation imposed by (b).

Lagrange points are special positions in a conﬁguration of two bodies in circular revolution around each other, e.g., the Earth-Moon system or the Sun-Earth system.

Gravity assists at the Moon or other planets are important to gain suﬃcient energy for interplanetary missions.

Orbits and missions: SSO, Lagrange points, gravity assist

Sun-Synchronous Orbits

Earth’s deviation from spherical symmetry is mainly due to its equatorial bulge caused by centrifugal action (which makes the planet an oblate ellipsoid rather than a perfect spheroid). For inclined (and not exactly polar) satellite orbits, this leads to a precession (i.e., a slow rotation) of the orbital plane. The precession rate matches the eﬀect of Earth’s revolution around the Sun for special choices of the

altitude: typically 600–800 km, and the

inclination: about 98◦_{(i.e., an almost polar orbit but slightly retrograde}

revolution with respect to the sense of the Earth’s rotation).

Then the orbital plane is ﬁxed with respect to the Sun which explains the term Sun-Synchronous Orbit.

Caution: Do not confuse with the term “Heliosynchronous Orbit”: that refers to a orbital motion around the Sun (!) with a period that matches the solar rotation period.

(6)

Sun-Synchronous Orbit (cont’d)

[NASA/Landsat (7)]

Advantages:

near-constant solar illumination

condi-tions of the satellite’s surface footprint,

satellite orientation with respect to the

Sun can be ﬁxed (e.g., for a constant energy supply by solar panels, solar and astronomical observations),

SSOs are typically used for

remote sensing of the Earth surface

(Landsat, ERS),

meteorological satellites, spy satellites,

observation of the Sun (TRACE,

Yohkoh, ACRIMSat, Hinode),

astronomical telescopes (IAS).

21

Lagrange points

[NASA/WMAP (8)]

If two celestial bodies are in circular orbits around each other, there are ﬁve positions where another (much less massive) object can co-rotate with the conﬁguration.

Spacecraft missions at Lagrange points

L1: SOHO, ACE.

L2: WMAP, and future space

observatories like Herschel, Planck, and Gaia.

L4 and L5: STEREO.

22

Gravity assists or Swing-bys:

= close ﬂybys at a planet to alter the velocity of a spacecraft.

Missions to the outer planets, e.g., Cassini to Saturn, make use

of gravity assists to gain speed.

[NASA (9)]

Gravity assists are also used to slow down space vehicles. Useful

for missions to the inner planets, e.g., MESSENGER to Mercury.

(7)

Gravity assist analogy: elastic collision

Light objects can change their speed substantially when they

bounce oﬀ a heavy object in motion.

[JPL/NASA (11)]

In head-on elastic collision be-tween a light object (mass m and

velocity v) and a heavy object

(mass M and velocity V ), the

post-collision velocity of the light object is given by v ₌ (m − M)v + 2MV m + M which simpliﬁes to v ₌ _{−v + 2V} in the limitm/M → 0. 25

Figures and references

(1) The illustrations of Kepler’s laws are taken from the

HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html (URL checked on 22 February 2008).

(2) Image credit: Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Image:Orbit1.svg, URL checked on 22 February 2008).

(3) Image credit: NASA (http://spaceflight.nasa.gov/gallery/, URL checked on 22 February 2008).

(4) Image credit: Geoforschungszentrum (GFZ)

Potsdam, CHAMP mission web pages

(http://www.gfz-potsdam.de/champ/systems/index SYSTEMS.html, URL checked on 22 February 2008).

(5) Image credit: Heavens Above web pages are developed and maintained by Chris Peat at http://www.heavens-above.com/ (URL checked on 23 February 2008). The diagram shows the orbit of Radionet-3 operated by Sirius Satellite Radio.

26

Orbits and missions: Figures and references Figures and references (cont’d)

(6) Image credit: Wikipedia Commons, http://commons.wikimedia.org/wiki/. File names of images: Orbits around earth scale diagram.svg, Molniya.jpg, Hohmann transfer orbit.svg (URLs checked on 23 February 2008).

(7) Image credit: NASA Landsat Handbook web pages (http://landsathandbook.gsfc.nasa.gov/handbook/handbook toc.html, URL checked on 23 February 2008).

(8) Image credit: NASA, public outreach pages of the WMAP mission (http://map.gsfc.nasa.gov/m mm/ob techorbit1.html, URL checked on 23 February 2008).

(9) Image credit: JPL/NASA, public outreach web page of the Cassini-Huygens mission to Saturn and Titan (http://cassini-huygens.jpl.nasa.gov/mission/gravity-assists.cfm, URL checked on 24 February 2008).

(10) Image taken from the MESSENGER web site at JHU/APL (http://messenger.jhuapl.edu/the mission/trajectory.html, URL checked on 24 February 2008).

(11) Cartoon taken from the the JPL web page A Gravity Assist Primer at http://www2.jpl.nasa.gov/basics/grav/primer.html (URL checked on 24 February 2008).

Review questions and further reading

Review questions

Explain the following key terms of celestial and orbital mechanics:

as-trodynamics, natural satellites, artiﬁcial satellites, revolution, planetary rotation.

Explain and discuss Kepler’s three law of planetary motion. Draw an

ellipse and explain the following terms: foci, semi-major and semi-minor axis, eccentricity. Which law determines the velocity along the orbit? How is the orbital period related with the semi-major axis? Which law allows you to work out the mass of the central body, and how?

What is the barycenter of a two-body system? Which form does Kepler’s

third law assume if the two bodies are equal in mass?

How is the pericenter (apocenter) of a Kepler orbit called if the central

body is (a) the Sun, (b) the Earth, and (c) the Moon?

Name and explain three of the six Keplerian orbital elements.

What is the altitude range of Low Earth Orbits (LEOs), and which kind

of satellites populate LEOs? What are the advantages and disadvantages of equatorial and polar LEOs?

(8)

Orbits and missions: Review questions and further reading Review questions (cont’d)

Deﬁne and discuss: Geosynchronous Orbit (GSO), Geostationary Orbit

(GEO), Geostationary Transfer Orbit (GTO).

What is the altitude range of Medium Earth Orbits (MEOs), and which

kind of satellites populate MEOs? What is a Semi-Synchronous Orbit?

What causes the precession of the orbital plane of satellites at

Sun-Synchronous Orbits (SSOs)? Which possibilities do SSOs oﬀer, and what kind of satellites are placed at SSOs?

What are Lagrange points, and which spacecraft missions take

advan-tage of them?

What is meant by a gravity assist? Which spacecraft missions make use

of this maneuvers and how? Textbooks

Celestial mechanics is covered in textbooks on classical mechanics. Orbital mechanics is also addressed in printed and electronic material

on spaceﬂight, rockets, and propulsion systems.

29

Orbits and missions: Review questions and further reading Web resources

The Jet Propulsion Laboratary maintains a suite of web pages that

provides a comprehensive introduction to the subjects discussed in this lecture. Visit Basics of Space Flight at http://www2.jpl.nasa.gov/basics/

David Stern’s educational web pages at http://www.phy6.org/ provide

information on a variety of space-related topics. In particular, have a look at the suite of pages entitled From Stargazers to Starships at http://www.phy6.org/stargaze/Sintro.htm.

HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html.

A public outreach programme on space physics is coordinated

at the Rice University (Project Manager: Patricia Reiﬀ), see http://earth.rice.edu/connected/spaceupdate.html. A very comprehen-sive list of links is http://space.rice.edu/ISTP/. A nice glossary can be found on the web site of the IMAGE spacecraft, see http://pluto.space.swri.edu/IMAGE/glossary/glossary intro.html.

The “Open House” web site of the Space Physics & Aeronomy section

of the American Geophysical Union also provides useful information, see http://www.windows.ucar.edu/openhouse/open house.html.

To reach the Oulu Space Physics Textbook, see http://www.oulu.fi/~spaceweb/textbook/.

30

Additional questions and problems

Problem 1

Compute the ﬁrst cosmic velocity. It is deﬁned as the (hypothetical) orbital velocity of an Earth satellite on a circular orbit at zero altitude, i.e., with semi-major axis a = 1 RE (one Earth radius).

Problem 2

How does the ﬁrst cosmic velocity relate to the escape velocity ? The latter is also called the second cosmic velocity.

Problem 3

Verify that the semi-major axis of a geosynchronous orbit (GSO) is_{a = 6.6 RE.} Problem 4

How long does it take to go from Earth to Mars by means of a Hohmann tranfer orbit? Where should Mars be located when the maneuver starts? (This is a somewhat longer exercise.)

Sample solutions of the problems

Sample solution of problem 1

On a circular orbit, the velocityV1 is given by the ratio of the circumference

of the circle and the orbital periodT , i.e., V1= 2πa/T , and therefore,

V2 1 =

4π2_a2

T2 .

Here a denotes the radius of the circle. We insert Kepler’s third law T2 ₌

4π2_a3_{/(GM) to obtain} V2 1 = 4π2_a2 4π2_a3GM = GM a and, ﬁnally, V1 =

GM/a. Inserting the values M = ME = 5.98 · 1024_kg,

G = 6.673 · 10−11_m3_kg−1_s−2_{, and} _{a = RE = 6.371 · 10}6_{m yields}

V1 = 7.9 km/s

(9)

Orbits and missions: Sample solutions Sample solution of problem 2 The escape velocity is given by

V2 = 2GM

a = √

2V1

and thus diﬀers from the ﬁrst cosmic velocity by a factor of√2 = 1.414.

We apply Kepler’s third law T2_{= 4}_π2_a3_{/(GM) with central body mass M =}

ME = 5.98·1024_{kg. The orbital period is given (}_{T = 23}h₅₆m_{= 86 160 s), and}

we solve for the semi-major axis a and to yield a m = 6.673 · 10−11_{· 5.98 · 10}24_{· (86160)}2 4π2 1/3 = 4.22 · 107 and a = 6.6 RE.

The solution of the problem can be found on the web page From Stargazers

to Starships – #21b Flight to Mars: How Long? Along what Path? by David

Stern: http://www-spof.gsfc.nasa.gov/stargaze/Smars1.htm.