Introduction to Astronautics Sissejuhatus kosmonautikasse

1. Tallinn University of Technology Introduction to AstronauticsSissejuhatus kosmonautikasse Vladislav Pustõnski 2009 – 2010

2. Orbital elements & types of orbits If we consider the satellite a point mass, six elements will be needed to describe its position and attitude, these are three coordinates and three components of its velocity vector (the seventh parameter is needed for time reference moment). However, this choice of elements is not the most convenient. Given that the motion of satellites (artificial as well as natural) is generally represented as Keplerian motion (with due account of perturbations), Keplerian elements are commonly used. Orbital elements 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 the orbit in its plane. These are: a – semi-major axis of the orbit, e – eccentricity of the orbit. 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 of reference. The equatorial plane of the planet is usually taken for the plane of reference. In the case of the Sun (and also the Moon!), the plane reference is the ecliptical plane. Once the plane of reference is defined, the following orbital elements are introduced: i – inclination,  – longitude of the ascending node.

3. Inclination is the angle between the orbital plane and the reference plane. Inclination is equal to the angle between the normal to the orbital plane and the plane of reference, the direction of the normal is determined by the right-hand rule. Zero inclination (i = 00) corresponds to equatorial orbits (which lie in the equatorial plane), i = 900 corresponds to polar orbits (lying in 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 node is the angle between the vernal point and the ascending node (north node). Let’s recall that the vernal point is the point on the celestial sphere where the Sun crosses the equatorial plane passing from the Southern hemisphere to the Northern hemisphere in spring. Analogically, the ascending node is the point at which the object moves north 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, is descending node (south node) and the line connecting these points is the line of nodes. Because of the precession of the Earth, the vernal point moves slowly along the ecliptic. By this 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 the object on the orbit:  – argument of periapsis, M0 – mean anomaly for epoh. Argument of periapsis is the angle between the line of apsides and the line of nodes measured in the orbital plane from the ascending node to the periapsis. The angle is measured in the direction of motion.

4. Mean anomaly for epoch is the mean anomaly measured from a chosen epocht0. It is defined by the relation P – period Mean anomaly at epoch enables to find true anomaly , this is the angle between the line of nodes and the radius-vector of the object. The relations between M0 and  are as follows E – eccentric anomaly Analogical relations keep for hyperbolic orbits. So, the full standard set of Keplerian elements is a – semi-major axis of the orbit, e – eccentricity of the orbit, i – inclination,  – longitude of the ascending node,  – argument of periapsis, M0 – mean anomaly for epoch. Epocht0is, however, the seventh element in this set. In aerospace engineering a slightly different set of elements is generally used. Epocht0is added explicitly, and mean motionn is introduced instead of semi-major axis. Mean motion is reciprocal of the period P, it is usually expressed in revolutions per day. The relation between mean motion and semi-major axis is given by Kepler’s Third Law.  – gravitational parameter 86 400 is the number of seconds per day

5. For Sun-orbiting satellites time of perihelion passage is used instead of mean anomaly for epoh – that is the moment when the object have passed its perihelion. However we should remember that Keplerian motion is idealization, since motion of real objects 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 elements give their values for a certain epoch. Orbital motion is close to Keplerian on short time intervals near this epoch. At longer intervals the orbital elements should be found with the aid of orbital mechanics.

6. Earth’s 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. Types of orbits Geocentric orbits We will start with geocentric orbits, i.e. orbits around the Earth. Role of inclination Let’s recall that inclination is the angle between the equator and the current orbital plane. Because of the Earth’s rotation, orbits with different inclinations have quite different shape of their ground tracks – projections of the orbit onto the Earth’s 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 equatorial orbit. In this case, the Earth’s 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 at both 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 R0 – radius of the Earth h – altitude of the orbit

7. For 200-km orbit,   140, for 2000-km orbit,   400, for 20 000-km orbit,   760. It is clear that polar regions are not observable even from very high equatorial orbits. The Earth’s oblateness, due to its symmetry, cannot change the inclination of equatorial orbits, 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 the orbital inclination. Since the Earth rotates, at each orbit the ground track shifts at a certain distance which depends on the orbital period. For instance, the orbital period of the Hubble Space 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 consecutive orbits the track shifts, finally covering the whole surface between the latitudes i deg North and i deg South (since precise proportionality between the orbital period and the Earth’s rotational period is impossible in practice, the track will not be closed). So, inclined orbits enable to observe wide areas on the surface, the northern most and southern most latitudes depending on the value of i. It depends on the orbital period, how often the satellite appears above this or that point on the surface. If P > 12h, the satellite moves more than 1800 to the West, so in fact it shifts to the East. If P = 24h (geosynchronous orbit, GEO), the Earth makes one full turn, thus at each orbit the satellite will fly above the same places on the surface and will never appear above the rest of the Earth. Due to the Kepler’s Third Law, all geosynchronous orbits have identical semi-major axes defined by the following relation  – gravitational parameter PEarth – period of the Earth’s rotation

8. Satellites having periods strictly proportional to the Earth’s rotational period, also pass over the same points on the surface (for instance, Molniya orbit with P = 12h, this is so-called semi-synchronous orbit since the period is about half of the period of the Earth’s 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 some latitude. 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 Earth’s 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 It is seen that a certain value of inclination i exists for which no shift is present. This value is i = arccos (1/5) =63.40. Molniya and Tundra orbits have that inclination. Due to perturbations from the Earth’s 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.

9. 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 orbits (see further) use this kind of perturbation. The case i = 900 corresponds to a polar orbit. 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 Earth’s rotation). So, a satellite in a polar orbit cover the whole Earth’s 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 satellites and 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 Earth’s 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 Earth’s surface (or one region) at the same local solar time. That means that at each passage the illumination angle is nearly the same.

10. 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 satellite’s 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-mentioned effect of orbital plane precession caused by the Earth’s 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 600 – 1000 km (to avoid rapid decay due to atmospheric drag), this corresponds to periods of about 90 – 100 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 satellite, the orbital plane of which remains near the terminator. The advantage of such orbit is that the satellite always “sees” the Sun (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.

11. 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 Earth’s 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 Earth’s rotation, but also because they enable to launch low inclination satellites with less penalty for inclination change maneuvers. This is quite important for launches of geostationary satellites (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 of launch windows, i.e. periods of time when the launch is possible.

12. 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); • 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); • 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.

13. The tasks of the satellite imply the altitude of its orbit. Orbits with very different altitudes are used. Orbital altitude Low Earth Orbits, LEO are 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 is usually 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 inner Van 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, ~ 90 – 130 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 sensing satellites 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.

14. 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, MEO are orbits with altitudes from the top of LEO to the geostationary orbits, i.e. 2000 km < h < 36 000 km. Orbital periods are 2 – 24 hours. These orbits are used generally by positioning and navigation satellites like GPS satellites, 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 Orbit, GSO is 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: – Earth’s gravitational parameter

15. A satellite placed into this orbit will have the identical angular velocity as the points on the Earth’s surface beneath it have due to the Earth’s 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 no need 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 3 in 1964. Because of perturbations from the Earth’s 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.

16. 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 ~2·36 000/300 000 =1/4 sec (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 International Telecommunication Union. 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 Graveyard Orbits, or Supersynchronous Orbits. Their height is ordinarily several hundreds of kilometers above GSO. Several m/s of characteristic velocity are required for the corresponding transfer maneuver. To reach GSO, satellites follow a so-called Geostationary Transfer Orbit (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 the launch 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.

17. 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 release their cargo not in the first, but in later apogees, one by one. This helps to place different payloads 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, and they decay quickly, in several years or less. High Earth Orbits, HEO are geocentric orbits with apogees above GSO. They are rare to use. The subset of them are Highly Elliptical Orbits. Apogees of Molniya & Tundra orbit lie above GSO, which label them as HEOs.

18. Other orbits Escape (or Parabolic) 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 provide the 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 the latter case escape occurs due to perturbations which increase the velocity to the hyperbolic. Hyperbolic Orbits. The orbit with eccentricity e > 1. Satellites leave their central bodies following 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 Orbit is 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 to perturbations or atmospheric drag. Heliocentric Orbits are orbits of space probes during their missions to objects of the Solar System, the Sun being in their focus. Some space probes (Pioneer 10/11, Voyagers, New Horizons) at the present moment move along Solar hyperbolic orbits. Halo Orbits is a special type of three-dimensional orbits near the Lagrangian points L1, L2 & L3. Lagrangian points (libration points) present in the system Earth-Sun and Earth-Moon are 5 points in the plane of the reciprocal orbit of two celestial bodies where a spacecraft may retain 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) are stable: after a small perturbation the spacecraft will continue to move in the vicinity of them.

19. They are also called Trojan points. Nearly stable orbits (quasi-periodic in the real n-body problem) exist near the unstable points, they are called halo orbits. A satellite at these orbits do not orbit the Lagrangian point, but moves in a closed path near it. These orbits are not exactly stable and need station-keeping maneuvers. The first object in a halo orbit was ISEE-3 joint 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.

20. End of the Lecture 4

21. Elements of elliptical orbit

22. Zero-inclination orbit, i = 00 Ground track Satellite on a zero-inclination circular orbit

23. Orbit of the Hubble Space Telescope, i = 28.50 Ground track HST orbit, i = 28.50

24. Tundra geosynchronous orbit, i 63.40 Tundra GSO of Sirius Satellite Radio Ground track

25. Ground track Molniya orbit, i 63.40 Molniya orbit

26. Rotation of the line of apsides Line of apsides rotates

27. Nodes regression Rotation of the line of nodes

28. Polar orbit, i 900 Ground track Orbit of a near-polar satellite

29. Sun-synchronous orbit Non-synchronous and synchronous orbits Landsat 7 Earth-imaging satellite orbit

30. Low Earth orbit Iridium constellation Voice & data transmission Altitude 780 km, Inclination 86.40 66 active satellites Concentration of satellites and space debris. LEO and GEO/GSO are clarely noticible

31. Geostationary orbit GSO satellite relative position Inmarsat global coverage Telephony & data

32. Lagrangian points Lagrangian points in the system Earth-Moon. Points L1, L2 & L3 are unstable, points L4 & L5 are stable

33. SOHO halo orbit Halo orbit of SOHO laboratory near L1 Lagrangian point between the Sun and the Earth