Astronautics Lecture4

7/30/2019 Astronautics Lecture4

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Introduction to Astronautics

Sissejuhatus kosmonautikasse

Vladislav Pustnski

20092010

Tallinn University of Technology


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Orbital elements & types of orbitsOrbital elementsIf we consider the satellite a point mass, six elements will be needed to describe its positionand attitude, these are three coordinates and three components of its velocity vector (theseventh parameter is needed for time reference moment). However, this choice of elements isnot the most convenient. Given that the motion of satellites (artificial as well as natural) isgenerally represented as Kepler ian motion(with due account of perturbations), Keplerianelementsare commonly used.Further we will consider a satellite of the Earth and describe its orbital elements in the Earth-relative frame. If the object moves along a near-Keplerian path close to another planet, a

natural satellite or a star, Keplerian elements are introduced analogically.Since Keplerian orbits are planar, the first pair of the elements refers to the parameters of theorbit in its plane. These are:

asemi-major axis of the orbi t,

eeccentr icity of the orbi t.

These elements define the size of the orbit (a) and its oblateness (e).

The second pair of the elements defines the orientation of the orbit relative to the plane ofreference. The equatorial plane of the planet is usually taken for the plane of reference. In thecase of the Sun (and also the Moon!), the plane reference is the ecliptical plane. Once theplane of reference is defined, the following orbital elements are introduced:

iinclination,

longitude of the ascending node.


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Inclinationis the angle between the orbital plane and the reference plane. Inclination is equalto the angle between the normal to the orbital plane and the plane of reference, the direction ofthe normal is determined by the right-hand rule. Zero inclination (i = 00) corresponds toequator ial orbits(which lie in the equatorial plane), i = 900 corresponds to polar orbits(lyingin a meridian plane and passing above the poles). i < 900 corresponds to direct (prograde)

motion, i > 900

corresponds to retrograde motion.Longitude of the ascending nodeis the angle between the vernal poin tand the ascendingnode(north node). Lets recall that the vernal point is the point on the celestial sphere wherethe Sun crosses the equatorial plane passing from the Southern hemisphere to the Northernhemisphere in spring. Analogically, the ascending node is the point at which the object movesnorth through the plane of reference, passing to the Northern hemisphere. The opposite point,

where the object crosses the plane of reference moving from the North to the South, isdescending node(south node) and the line connecting these points is the l ine of nodes.

Because of the precession of the Earth, the vernal point moves slowly along the ecliptic. Bythis reason the epoch of the vernal point should also be set.

Two last elements define the position of the orbit in its plane and the current position of theobject on the orbit:

argument of per iapsis,M0mean anomaly for epoh.

Argument of per iapsisis the angle between the line of apsides and the line of nodes measuredin the orbital plane from the ascending node to the periapsis. The angle is measured in thedirection of motion.


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EeEM sin0

2tan

1

1

2tan

E

e

e

Mean anomaly at epoch enables to findtrue anomaly, this is the angle betweenthe line of nodes and the radius-vector ofthe object. The relations between M0and are as follows

Eeccentr ic anomaly

Analogical relations keep for hyperbolic orbits.

So, the fullstandardset of Keplerian elements is

asemi-major axis of the orbi t,

eeccentr icity of the orbi t,

iinclination,

longitude of the ascending node,argument of per iapsis,M0mean anomaly for epoch.

Epocht0is, however, the seventh element in this set.

Mean anomaly for epochis the mean anomalymeasured from a chosen epocht0. It is defined bythe relation

P

ttM 0

02

Pperiod

In aerospace engineering a slightly different set of elementsis generally used. Epocht0is added explicitly, and meanmotionnis introduced instead of semi-major axis. Meanmotionis reciprocal of the periodP, it is usually expressedin revolutions per day. The relation between mean motion

and semi-major axis is given byKeplers Third Law.

32

40086

an

gravitational parameter86 400 is the number of

seconds per day


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For Sun-orbiting satellites time of per iheli on passageis used instead ofmean anomaly forepohthat is the moment when the object have passed its perihelion.

However we should remember that Keplerian motion is idealization, since motion of realobjects is always perturbed by non-central forces. Thus, the orbital elements are not constant,

they change with time and should be regarded as time functions. Tables of orbital elementsgive their values for a certain epoch. Orbital motion is close to Keplerian on short timeintervals near this epoch. At longer intervals the orbital elements should be found with the aidof orbital mechanics.


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Types of orbits

Role of inclination

Earths satellites perform different tasks, and the task of the satellite defines its orbit. Certain

orbits are better for certain purposes, so for every spacecraft a suitable orbit should be

provided. Because of limitations of orbital mechanics and some other reasons discussed

further, positions of launch sites also play important role in formation of orbits.

Lets recall that inclinationis the angle between the equator and the current orbital plane.Because of the Earths rotation, orbits with different inclinations have quite different shape of

theirground tracksprojections of the orbit onto the Earths surface (along the ground track

the satellite is in the zenith.

If the inclination of the satellite is i = 00, the satellite will remain above the equator. This is

equator ial orbit. In this case, the Earths rotation will not bring new zones but equatorial

beneath the ground track. If the orbital altitude is small, only a narrow band of the surface atboth sides of the equator is visible from the satellite (and, of course, vice versa: the satellite

would be seen by a terrestrial observer only from this region). The angular semi-width of this

band (from its center to its edge) may be computed from the obvious geometric relation

hR

R

0

0arccos

R0radius of the Earth

haltitude of the orbit

Geocentric orbitsWe will start with geocentric orbits, i.e. orbits around the Earth.


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For 200-km orbit, 140, for 2000-km orbit, 400, for 20 000-km orbit, 760. It isclear that polar regions are not observable even from very high equatorial orbits.

The Earths oblateness, due to its symmetry, cannot change the inclination of equatorialorbits, so their inclination may only change due to solar and lunar perturbations.

In the case 00 < i < 900 the orbit enters higher latitudes, the highest latitude being equal to theorbital inclination. Since the Earth rotates, at each orbit the ground track shifts at a certaindistance which depends on the orbital period. For instance, the orbital period of the HubbleSpace Telescope is ~96 min, i.e. ~0.07d. So, at each orbit the Earth shifts beneath HST by 0.07 3600 250 to the East, that corresponds to HST shift by 250 to the West. At consecutiveorbits the track shifts, finally covering the whole surface between the latitudes i deg North and

ideg South (since precise proportionality between the orbital period and the Earths rotationalperiod is impossible in practice, the track will not be closed). So, inclined orbits enable toobserve wide areas on the surface, the northern most and southern most latitudes dependingon the value ofi.

It depends on the orbital period, how often the satelliteappears above this or that point on the surface. IfP> 12h,the satellite moves more than 1800 to the West, so in fact itshifts to the East. IfP= 24h (geosynchronous orbit, GEO),the Earth makes one full turn, thus at each orbit the satellitewill fly above the same places on the surface and will neverappear above the rest of the Earth. Due to the KeplersThird Law, all geosynchronous orbits have identical semi-

major axes defined by the following relation

3

2

2

4 EarthP

a

gravitational parameter

PEarth

period of the Earthsrotation


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Satellites having periods strictly proportional to the Earths rotational period, also pass over

the same points on the surface (for instance, Molniya orbitwithP= 12h, this is so-called

semi-synchr onous orbitsince the period is about half of the period of the Earths rotation).

The shape of the ground track depends on period and eccentricity of the orbit. Since the

surface in intermediate latitudes moves slower than at the equator, a long-period satellite that

moves slower than the surface at the equator may move quicker than the surface at somelatitude. Such satellite changes its direction of motion from retrograde to direct (prograde) and

moves from the East to the West near equatorial latitudes. The ground track of such satellite

may even intersect itself at one orbit. The same situation takes place also for highly elliptic

orbits (see Tundra & Molniya orbits), at which satellites move very slowly near the apogee

and quickly near the perigee.

An important perturbation caused by the Earths oblateness and influencing orientation of

inclined (and not only inclined) elliptic orbits is rotation of argument of perigee (rotation of

the line of apsides). It would shift the initial near-polar position of the apogee to the

equatorial latitudes. However, this rotation is eccentricity-dependent. The corresponding

relation for the daily change is

Due to perturbations from the Earths oblateness, the longitude of the ascending node of

inclined orbits constantly changes (as well as other elements). For direct satellites this means

westward rotation of the ascending node. This phenomenon, caused by precession of the

orbital plane, is called nodes regression.

222

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00

1

1cos55

e

i

a

Rday

It is seen that a certain value of inclination i exists for whichno shift is present. This value is i = arccos (1/5) =63.40.

Molniya and Tundra orbits have that inclination.


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For retrograde satellites the rotation occurs in the opposite direction. For low near-

equatorial orbits the regression is several degrees per day. The influence of the oblateness

quickly drops with altitude. For polar and near-polar orbits this perturbation is also low

due to the symmetry of the equatorial bulge. The rotation of the nodal line (as well as

other perturbations) should be taken into the account during the mission planning. Sun-

synchronous orbi ts(see further) use this kind of perturbation.

The case i = 900 corresponds to a polar orbi t. This orbit passes above the poles (or very

close to the poles). Except the special case of the polar geosynchronous orbit, a polar

satellite at such orbit passes the equator at different longitudes at each orbit (because of

the Earths rotation). So, a satellite in a polar orbit cover the whole Earths surface,

visiting each point of the planet (the period of consecutive visits depends on the semi-

major axis and the eccentricity). This property makes polar orbits very suitable for

weather satel l i tesand reconnaissance orbiters, as well as for other satellites whose task

is observations of the surface (mapping etc.).

It should be taken into account that due to the Earths rotation polar satellites at each

latitude have western velocity component. That means that to launch a polar satellite, one

should not chose exactly southern or northern azimuth, but should launch at a (small)angle with the meridian.

A variation of the near-polar orbit is a Sun-synchronous orbit, SSO. Satellites at these orbits

pass over regions on the Earths surface (or one region) at the same local solar time. That

means that at each passage the illumination angle is nearly the same.


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This is important for remote weather sensing (for instance, of temperature), when long-term

changes are tracked; in such cases changes with the local time should be eliminated.

Since the Earth orbits the Sun, the local time of the satellites passage would change with

seasons (2 hours per month): the Sun moves eastward. So the orbital plane of the satellite

should rotate with the Sun to follow its motion. This is provided by the above-mentionedeffect of orbital plane precession caused by the Earths oblateness.

Since the orbital plane should rotate (slowly) eastward following the Sun, the orbit should be

(slightly) retrograde. This is provided by near-polar retrograde orbits. A typical sun-

synchronous satellite has the orbital height of about 6001000 km (to avoid rapid decay due

to atmospheric drag), this corresponds to periods of about 90100 min. The inclination

providing such nodes regression is i

980

.Some variations of solar-synchronous orbits exist. For instance, for highly elliptic orbits the

fixed solar time of passage occurs only over one point on the surface (typically the perigee).

For instance, a 96-min satellite will make exactly 15 orbits per day, visiting 15 different

longitudes at the same solar time in each location; on the next day it will begin the round

with the first location. The special case is a dawn/dusk satell i te, the orbital plane of which

remains near the terminator. The advantage of such orbit is that the satellite always sees theSun (it may be useful for continuous solar observations), and that the Sun always illuminates

its solar cells, which are never shadowed by the Earth.

For higher altitude the inclination angle required for such satellite becomes smaller, so it will

not be able to see polar regions (and, of course, it will be farther from the observed objects).

Thus sun-synchronous satellites usually do not have very long semi-major axes.


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It is important to notice that since low-inclination satellites do not pass over higher latitudes,

it is impossible to launch them from the launch sites in higher latitudes without changing the

orbital inclination of the satellite during or after launch. If the latitude of the launch site is ,

the minimum inclination that may be provided without change is i = , this is done by direct

launch to the East, at the same time the maximum velocity gain from the Earths rotation is

obtained. (Of course, retrograde launches to latitudes > 1800 - i are also possible). That

means that launches to all latitudes are possible only from equatorial launch sites. The higher

is the latitude of the launch site, more significant orbit inclination changes are needed to

launch equatorial and near-equatorial satellites. However, as we will see later, change of

inclination is quite an expensive maneuver, in terms of characteristic velocity. That means

that near-equatorial launch sites have great advantage not only because of the higher

characteristic velocity gain due to the Earths rotation, but also because they enable to launch

low inclination satellites with less penalty for inclination change maneuvers. This is quite

important for launches ofgeostationary satel l i tes(see further) that have zero inclination. Just

for this reason, for instance, Europe established its launch site in Kourou (latitude = 503)

and the Sea Launch performs Zenit-3SL launches from equatorial zone (00N 1540W) in the

Pacific. This permits to increase the payload of Zenit-3SL rocket to GEO by about 15% -20% per cent compared with the launch from Cape Canaveral ( = 28030).

From the considerations of launching the payload more effectively, it is always desirable to

launch directly into the orbital plane needed. This limits the number oflaunch windows, i.e.

periods of time when the launch is possible.


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1) If the inclination of the desired orbital plane is higher than the latitude of the launch

site, there are two launch windows per day, at the moments than the launch site crosses

the orbital plane (two launch times, the gap between them is 12 hours; the azimuths

differ by the same angle from the East-West direction);

2) If the inclination of the desired orbital plane is equal than the latitude of the launch site,

there is one launch window per day (launch directly to the East to the direct orbit / West

to the retrograde orbit);

3) If the inclination of the desired orbital plane is lower than the latitude of the launch site,

no launch window exist for a direct launch into the desired orbital plane, and maneuvers

to change of the inclination are unavoidable.


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Orbital alti tudeThe tasks of the satellite imply the altitude of its orbit. Orbits with very different altitudes areused.

Low Earth Orbits, LEOare orbits with altitudes < 2000 km. Since objects with orbital height

of ~ 200 km and lower rapidly decay due to atmospheric drag, the lower limit of LEO isusually adopted to be 160 km. Because of its low altitude, LEO need smaller characteristic

velocities to achieve than higher orbits. It also lies mostly below the innerVan Allen

radiation belt(where high concentrations of energetic particles exist). This is why orbits of

most crewed spacecraft are in LEO (with the exception of the Apollo flights to the Moon).

The record altitude was the apogee of Gemini XI, about 1370 km. Orbital periods in LEO are

the smallest, ~ 90130 min. This enables short surface locations revisiting times, specially

for near-equatorial satellites.

Although GSO is needed for many communication applications, LEO are still widely used

for the same purposes since they are much simpler to reach. However, to provide continuous

coverage, a constellation of satellites at LEO is needed for communication purposes. Due to

low height a satellite covers only a small area on the surface, usually < 4000 km. Satellites

move continuously relative to the surface, remaining above the local horizon of an observer

only for a short period, usually < 20 min. So, constant relay between satellites is needed.

Remote sensingsatellites are also placed generally to LEO to be not too far from the objects

they observe; most of sun-synchronous orbits (with heights of ~800 km) are also in LEO.

Thus, LEO is the most populated region, about 10 000 objects with sizes > 10 cm are tracked.


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These are active and inactive satellites, spent upper stages of rockets, launch debris etc. In

LEO objects reside in the upper atmospheric layers and are subject to significant atmospheric

drag which leads to orbit decay. That makes it necessary to reboost active satellites from time

to time and thus, to spend additional propellant to keep them in orbit. For instance,

International Space Station is reboosted several times each year. Atmospheric drag strongly

depends on the solar activity (since density of the upper atmosphere may experience changes

of tens of per cent), so exact times of orbital decay are difficult to predict in long time-scales.

For instance, Skylab and Salyut-7 space stations decayed quicker when expected due to

increased solar activity.

Medium Earth Orbits, MEOare orbits with altitudes from the top of LEO to the

geostationary orbits, i.e. 2000 km < h < 36 000 km. Orbital periods are 224 hours. These

orbits are used generally by positioning and navigation satellites like GPSsatellites, and also

communication satellites covering the Poles. Atmospheric drag at these altitudes is very low

and lifetimes of these objects are very long. However, MEO lie inside van Allen radiation

belts where high concentrations of energetic charged particles exist, so long stay inside them

or frequent passage through them decrease the active lifetime of spacecraft due to

vulnerability of electronic components; shielding is needed to increase their lifetime.Geostationary Orbi t, GSOis the special case of GEO, it is the circular equatorial orbit with

the period equal to the rotation period of the Earth. From the formula for period we may

obtain the altitude of this orbit:

mRP

h 632

2

14

03

2

2

109.35104.64

86400104

4

6

Earths gravitational

parameter


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A satellite placed into this orbit will have the identical angular velocity as the points on the

Earths surface beneath it have due to the Earths rotation. Thus, the satellite will remain

steady relative to the surface of the Earth. This property is very useful for communication

satellites, since they have a fixed position for the terrestrial observer on this orbit. So a

satellite may keep its receivers and transmitters fixed to a service areas, and on the Earth one

may fix his receiver/transmitter to the certain point on the celestial sphere. Thus, there is noneed to redirect antennas following the orbital motion of the satellite. High altitude ensures

wide coverage: the area of visibility of a GSO satellite spreads till very high latitudes (~800).

The first satellite placed to this orbit was Syncom 3in 1964.

Because of perturbations from the Earths oblateness, from the Moon and the Sun, satellites

in GSO cannot stay exactly above one point on the surface but tend to drift from their

positions. This makes it necessary to correct their positions continuously (stationkeeping).Without corrections, the orbital inclination of a GSO satellite would change about one degree

per year (this is the mayor perturbation; satellites also have a minor tend to drift westward or

eastward). Each year about 50 m/s of characteristic velocity is required to keep a satellite at

its position, so it has to spend propellant. The amount of propellant onboard is generally the

main factor limiting lifetime of GSO satellites. Since at no moment the satellite is exactly

stationary, in fact all satellites in GSO are (nearly) geosynchronous, their tracks on the

surface are Lissajous figures.

There are several problems related to the use of GSO.

First, GSO is quite high and needs powerful boosters to achieve. So LEO communication

satellites are also used, but they do not possess the main advantage of GSO satellites.


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Second, it is problematic to use GSO satellites for areas located in high latitudes, i.e. near the

Poles, since satellites are close to the horizon or beneath it. This occurs in northern regions of

Russia, Canada, Greenland, in Arctic and Antarctic. The solution is to use geosynchronous or

semi-synchronous orbits, like Molniya or Tundra orbits.

Third, because of the large distance certain time delay exists in signal travel time, since the

speed of light is limited. The minimal delay is ~236 000/300 000 =1/4sec (distance traveled

in both ways). This may be important, for instance, in voice communications.

Forth, the length of this orbit is limited, so countries have to compete for slots in GSO. Slots

are allocated by the I nternational Telecommunication Union.

To reach GSO, satellites follow a so-called Geostationary Transfer Orbi t (GTO)after launch.

It is an elliptic orbit with the apogee laying at the equatorial plane and the height equal to the

height of GSO (~36 000 km). Perigee of GTO is low, it may be in LEO (for instance, if thelaunch is performed from the Space Shuttle with the use of a dedicated boost stage) or even

sometimes beneath the surface (if the satellite is sent to GTO by the upper stage of its launch

vehicle). Hohmann transfer (analyzed further) is generally applied. Although some launch

vehicles are able to place their payload directly to GSO, in most cases they leave their payload

in GTO, and satellites use their own motors to pass from GTO to GSO.

To avoid pollution of GSO with satellites out of service, they are transferred to storage orbits

in the end of their life, when their propellant comes to end. These are so-called GraveyardOrbits, orSupersynchronous Orbi ts. Their height is ordinarily several hundreds of

kilometers above GSO. Several m/s of characteristic velocity are required for the

corresponding transfer maneuver.


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This is done to increase the mass of the payload (there is no need to spend propellant to

transfer to GSO the heavy upper stage of the launch vehicle, and GSO satellites have engine

and propellant anyway to keep their position). A semi-major axis of a typical GTO is 24 500

km. Since most of launches occur from non-zero latitudes (except for the Sea Launch),

satellites need to change their inclination to pass to GSO. Change of inclination requires high

characteristic velocities, and this operation, as we shall see later, may be done near the

apogee (where the velocity is minimal) with much smaller characteristic velocity expense

than near the perigee. This is why change of inclination and orbital circularization are usually

performed by a single engine burn. Sometimes the satellite is sent to a GTO with a apogee

even higher than the GSO height to decrease the cost of the inclination change maneuver.

The inclination change maneuver is produced at this higher apogee, and then the apogee is

lowered to the GSO height; this double maneuver with an increased initial apogee may be

cheaper than change of inclination at the GSO height. Proximity of the launch site to the

equator highly favors passing from GTO to GSO.

Since now launch vehicles often carry multiple payloads to GSO, sometimes they releasetheir cargo not in the first, but in later apogees, one by one. This helps to place differentpayloads to different positions on GSO. The spent stages of launch vehicles remain on GTO,but their lifetime is short, since their mass/area ratio is small, their perigee height is low, andthey decay quickly, in several years or less.

High Earth Orbits, HEOare geocentric orbits with apogees above GSO. They are rare to

use. The subset of them are H ighly Ell iptical Orbits. Apogees of Molniya & Tundra orbit lie

above GSO, which label them as HEOs.


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Other orbitsEscape (or Parabol ic) Orbits. This is the ideal case of the object having the escape velocity.In reality the exact parabola is never present due to perturbations and impossibility to providethe exact value of velocity. In reality, objects may leave the central body having the

hyperbolic velocity (most cases) or a velocity slightly lower than the escape velocity; in thelatter case escape occurs due to perturbations which increase the velocity to the hyperbolic.

Hyperboli c Orbits. The orbit with eccentricity e > 1. Satellites leave their central bodiesfollowing this orbit. Deep space probes leaving the Earth are put to this orbit by their boosters,fly-by space probes also fly pass their target planets following hyperbolic trajectories.

Capture Orbi tis a reverse hyperbolic orbit, when a space probe approaches a planet with a

hyperbolic velocity, but is captured to a planetocentric orbit (or to a landing trajectory) due toperturbations or atmospheric drag.

Heli ocentri c Orbitsare orbits of space probes during their missions to objects of the SolarSystem, the Sun being in their focus. Some space probes (Pioneer 10/11, Voyagers, NewHorizons) at the present moment move along Solar hyperboli c orbits.

Halo Orbitsis a special type of three-dimensional orbits near the Lagrangian points L1, L2

& L3. Lagrangian points (li bration points) present in the system Earth-Sun and Earth-Moonare 5 points in the plane of the reciprocal orbit of two celestial bodies where a spacecraft mayretain a stationary position (in the absence of perturbations).

Points L1, L2 & L3 are unstable, a spacecraft will leave them after a minor perturbation.Points L4 & L5 (their angular distance from the line connecting the celestial bodies is 600) arestable: after a small perturbation the spacecraft will continue to move in the vicinity of them.


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They are also called Trojan poin ts. Nearly stable orbits (quasi-periodic in the real n-bodyproblem) exist near the unstable points, they are called halo orbits. A satellite at these orbitsdo not orbit the Lagrangian point, but moves in a closed path near it. These orbits are notexactly stable and need station-keeping maneuvers. The first object in a halo orbit was ISEE-3joint NASA/ESA solar mission in 1978, it was placed near the L1 Sun-Earth Lagrangian

point. SOHO solar laboratory was put into a similar orbit in 1996.Orbits around L1, L2 & L3 points are stable, their shape is Lissajous figures.


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End of the Lecture 4


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Elements of elliptical orbit


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Satellite on a zero-inclination

circular orbit

Zero-inclination orbit, i = 00

Ground track

O bit f th H bbl S T l i


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HST orbit, i = 28.50

Orbit of the Hubble Space Telescope, i =

28.50

Ground track


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Tundra GSO of Sirius Satellite

Radio

Tundra geosynchronous orbit, i63.40

Ground track


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Molniya orbit

Molniya orbit, i63.40

Ground track


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Line of apsides rotates

Rotation of the line of apsides


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Rotation of the line of nodes

Nodes regression


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Ground track

Polar orbit, i900

Orbit of a near-

polar satellite


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Non-synchronous

and synchronousorbits

Sun-synchronous orbit

Landsat 7 Earth-imaging

satellite orbit


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Iridium constellation

Voice & data transmissionAltitude 780 km,

Inclination 86.40

66 active satellites

Low Earth orbit

Concentration of satellites

and space debris. LEO and

GEO/GSO are clarely

noticible


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GSO satellite

relative position

Geostationary orbit

Inmarsat global coverage

Telephony & data

Lagrangian points


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Lagrangian points

Lagrangian points in the system Earth-

Moon. Points L1, L2 & L3 are unstable,

points L4 & L5 are stable


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Halo orbit of SOHO

laboratory near L1Lagrangian point

between the Sun and

the Earth

SOHO halo orbit

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