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Tallinn University of Technology. Introduction to Astronautics Sissejuhatus kosmonautikasse. Vladislav Pust õnski 2009 – 2010.

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orbital perturbations

In the first approximation, the motion of satellites is regarded as Keplerian, i.e. as the motion in the central field of gravity force. However, due to perturbations of different nature the real motion is not Keplerian. In practice, the methods of orbital mechanics consider this motion as a perturbed Keplerian motion. That means that in addition to the primary gravitational acceleration, the satellite experiences smaller perturbational accelerations, so the elements of its Keplerian orbit change with time, being time functions.

The principal sources of perturbations are atmospheric drag, the Earth’s oblateness, solar and lunar gravitational pull and solar radiation pressure. Perturbations may also be caused by the satellite itself, for instance, by minor escape of gases from the systems or uneven heat radiation emission. However, we will examine in detail only external sources of perturbations.

The role of different factors at the altitudes ~ 100 – 2000 km is illustrated by the Figure.

It is seen that at low altitudes atmospheric drag dominates, at several hundreds of kilometers the role of the Earth’s oblateness becomes important, solar and lunar perturbations also play their role. However one should remember that while other sources introduce mostly periodic perturbations, the atmospheric drag introduces a secular perturbation leading to continuous decay of the orbit. This is why this perturbation is quite important up to heights of about 1000 km, since the lifetime of objects at these heights is defined mainly by atmospheric drag (and vary from days to tens of years).

At higher orbits the influence of atmospheric drag decreases rapidly. The role of the Earth’s oblateness decreases as well, but slower. The influence of the Sun and the Moon increases, that of the Moon quicker. Role of other planets is much smaller: perturbations from the Jupiter are ~1% of lunisolar perturbations. Let us consider in detail the most important factors.

Orbital perturbations
atmospheric drag

Satellites in space near the Earth move through the rarefied layers of the upper atmosphere (so-called thermosphere), and are subject to atmospheric drag force directed opposite to the velocity vector. This force is represented by the equation

Atmospheric drag

A – area of the cross-section perpendicular to the velocity vector

 – local gas density

Cd – drag coefficient

Typical densities of the gas where satellites move are so low that flow is molecular (mean free path is very large), thus typical values of the drag coefficient Cd are 1.8 – 2.5. For hypersonic flow in the lower layers at re-entry the drag coefficient is Cd 1. However, we will analyze re-entry later in this course, here we only consider the orbital movement of the satellite before the final re-entry. Computation of the drag coefficient Cd for a specific object presents certain problems, since satellites have complex shapes. In the general case, atmospheric drag causes also appearance of the lift force in the direction perpendicular to the velocity vector; the formula for the lift force is the same as for the drag force with substitution of the drag coefficient Cd by the lift coefficientCL. However, the lift force should be taken into account only if very precise predictions are needed, so we will neglect it in our qualitative description.

The acceleration of the satellite under the influence of the drag force is evidently as follows

m – mass of the satellite.

m/CdA is called ballistic coefficient (BC), so the acceleration is

slide4

Objects with higher ballistic coefficient experience smaller accelerations. The lower is the mass and the larger is the cross-section area of a satellite, the stronger is atmospheric drag. For a typical 2-m round satellite with a mass of

2 tons, BC  2000/(2.2·3.14·12)  300 kg/m2. For small satellites ballistic coefficients are smaller (and atmospheric drag is more pronounced) since the area is proportional to square of linear dimensions and the mass is proportional to their cube. BC is small also for spent upper stages of launch vehicles due to their large size and moderate weight of empty tanks.

Atmospheric density quickly drops with altitude. For rough approximations, barometric formulae with varying coefficients may be used to compute density  at the height h:

H is the height scale. However, these are only approximate values, and they may significantly differ from the real values.

The uppermost atmosphere layers are very sensitive the space environment: the solar activity, the seasons, diurnal changes (actual values may differ several times from the mean). Thus, the drag acceleration is also sensitive to these factors. But since the general trend of density is a rapid decrease with the height, the drag acceleration also decreases rapidly. For a satellite with BC = 300 kg/m2 with the above-mentioned formulae we get adrag  1.7·10-4m/sec2 at the height of 200 km, and adrag  8.9·10-6m/sec2 at the height of 400 km. These values also define characteristic velocity needed to keep the object on the circular orbit. To keep the satellite on a 200-km orbit during 1 year, the characteristic velocity of Vch1.7·10-4 ·3.2·107  5400 m/s is needed, which is very high value. For a 400-km orbit, it is only Vch  300 m/s.

One may also compute that at the height of 100 km the drag acceleration is ~ 0.2 m/sec2, and during the time of one orbital period (P 5400 sec) the deceleration will be

slide5

~ 0.2·5400  1100 m/sec. That means definitive loss of the orbital velocity and actual re-entry. Thus, satellite cannot leave on stable orbits with heights about 130 km or lower.

As we shall see later, atmospheric drag leads to decrease of the orbital eccentricity, circularizing the orbit. Let us first examine evolution of a circular orbit in the atmosphere. The change of the semi-major axis per revolution may be estimated from the following considerations. The total energy of the satellite on a circular orbit with the radius (semi-major axis) a is the sum of its kinetic and potential energy:

 – gravitational parameter

ms – mass of the satellite

Vcirc – circular velocity

The relation between a small change of the semi-major axis a and the respective change of energy E is found from the derivative

The energy decreases because of the work of the drag force: E = - Adrag, and this work is the product of the drag force and the path of one revolution, which is 2a. So we get

Combining the beginning and the end and using the ballistic coefficient, we readily get

Analogically, the changes of the the period and the orbital velocity per one orbit may be estimated by the formulae

slide6

Note that while the semi-major axis and the period decrease, the velocity increases due to atmospheric drag, although the drag force is directed oppositely to the velocity. This is the so-called satellite paradox – the velocity of the satellite increases although it experience the frictional force. The explanation of this paradox is simple. The satellite moves towards the Earth along an inclined path, and there are two tangential forces in fact: atmospheric drag backwards and the tangential projection of the gravity force directed forward. The vector sum of these forces is directed forward, accelerating the satellite. From the point of view of orbital mechanics, the satellite that passes to a lower (quasi)circular orbit should have higher orbital velocity at this orbit, and that is what really happens.

The number of orbits before re-entry in the height range 150 km < h < 350 km may be very roughly estimated as

Multiplying this by the mean period, one may also estimate the lifetime T before the re-entry. Let us make these rough estimations for two heights, 200 km and 350 km. We will get for h = 200 km

arev -1.5 km, Prev -2 sec, Vrev 1 m/sec, n 20, T  30 hours. For h = 350 km:

arev -0.2 km, Prev -0.2 sec, Vrev 0.1 m/sec, n 1200, T  70 days.

It is important to remember that these numbers are rough estimates only since they depend on the atmospheric conditions and on the shape of the satellite.

The Earth’s atmosphere rotates with the Earth (with the linear velocity of about 460 m/s on the equator and slower on the higher latitudes), so the angle between the velocity vector of the satellite and the flow velocity vector is not zero in the general case. This leads to appearance of a lateral force, which changes the inclination of the orbit and the position of the line of nodes. In general, the inclination tends to decrease and the line of nodes tends to regress.

slide7

If the orbit is eccentric, atmospheric drag leads to orbital circularization. The reason is the following. Since the density of the atmosphere drops exponentially, the braking in the perigee for a highly elliptical orbit is much stronger in the apogee (also because the braking force is proportional to the square of velocity, and the velocity in the perigee is higher). Thus, for highly elliptical orbits it is possible to consider braking only near the perigee and to ignore at all braking in the apogee. Passing its perigee, the satellite experiences a short braking episode. The resulting decrease of the perigee is small. (If, in the first approximation, we consider the perigee braking as momentary, there will be no decrease at all.) However, due to reduction of the perigee velocity, the apogee altitude will change. It is simply to make sure that this change is proportional to the radius-vector of the apogee, so the higher is the apogee, more significant will be its decrease. But braking in the apogee is negligible for highly elliptic orbits, thus passing the apogee nearly do not influence the perigee. Thus, the perigee will decrease slowly and the apogee will decrease quickly (but slower and slower as the orbit approaches to circle).

The change of eccentricity per orbit in the first approximation is given by the formula

The formulae for the changes of semi-major axis, period and velocity remain the same in the first approximation, the value of atmospheric density should be taken near the perigee.

Predictions of the lifetime are quite complicated and not precise for high orbits, and they become more precise for short-term predictions (for lifetimes of several days it is often possible to predict the number of orbits before re-entry with an error of one or two orbits). The following issues are the principle sources influencing atmospheric conditions and so, the precision of predictions.

slide8

Diurnal variations. The density maximum follows the Sun but has a certain lag by 2 or 3 hours. At the equinoxes its center is on the equator, and moves to higher latitudes in other seasons. It is followed by the minimum in about 12 hours.

  • 27-day sun-rotation cycle. Active zones on the solar surface induce atmospheric density fluctuations correlating with the solar rotational cycle.
  • 11-year solar cycle. Solar activity influences atmospheric conditions, but with a lag of few years. Amplitudes vary as solar activity maxima vary from cycle to cycle.
  • Rotating atmosphere. Time variations due to the atmosphere differential rotation (at different heights the atmosphere is carried in a varying extent by the Earth’s surface).

The history of space flights knows several episodes when wrong estimations of drag led to undesired consequences. The US space station Skylab lacked engine for de-orbit maneuver, it was re-boosted by the last expedition in 1973 to a ~ 400 km orbit and was expected to stay in space waiting for arrival of the first Shuttle in the end of 1970s. The Shuttle should have docked a boosting device to the Skylab. But the readiness of the Shuttle delayed, and unexpected growth of solar activity led to more intense decay of the orbit (orientation of the unguided station was also estimated inexactly, some gravity effects were not taken into account, thus the estimated ballistic coefficient was not very accurate). On the last orbits of the Skylab in 1979, the mission control succeeded to change its orientation using the on-board gyros, thus reducing drag and extending the final orbit by several thousands of kilometers. This helped to avoid re-entry above North America, and most of debris fell in the Indian Ocean (some debris fell in Australia). The Russian space station Salyut-7 also re-entered earlier than predicted above the South America because of the increased solar activity (failed control system rendered it impossible to perform the re-entry maneuver).

earth oblateness

The Earth’s shape is not exact sphere. The polar radius is Rpol = 6357 km, and the mean equatorial radius is Req = 6378 km, thus the difference is 21 km. That means that the gravity field of the Earth is not central. The oblateness of the Earth leads to appearance of a non-central gravity force influencing the orbital motion of satellites. Additionally, the shape of the Earth has some other minor irregularities also influencing the orbital motion.

The general method applied in celestial mechanics to analysis of such fields is their decomposition into spherical harmonics. Thus, gravity potential is represented in the following form:

Earth oblateness

Zonal harmonics Jn consider the latitudinal deformations (deviations from the sphere), sectorial and tesseral harmonics consider longitudional deformations.

The values of the first harmonics are:

 – gravitational parameter

r – distance from the gravity center

Jn – zonal harmonics

Cnm, Snm– sectorial harmonics (if n = m) or tesseral harmonics (otherwise)

Pnm – Legendre polynomial

 – declination (latitude)

 – longitude

Other harmonics are smaller.

slide10

It is clearly seen that the major role plays the zonal harmonic J2, being 3 orders of magnitude larger than other harmonics. This harmonic corresponds to the oblateness of the Earth in the polar direction (J3 is responsible for the pear-shapedness). So, in the first approximation we may neglect all other summands and consider only this term. That leads us to the formula

The oblateness may be regarded as presence of an equatorial “bulge” on the generally spherical figure of the Earth. This bulge pulls the satellite towards the

equator, accelerating it on its way to lower latitudes and decelerating on its way to higher latitudes. The influence of the additional non-central potential on the orbital elements may be found with the aid of the perturbation theory, as it is usually done in celestial mechanics. The principle results are given below.

It turns out that in the first approximation the semi-major axis a and the eccentricity e of the orbit do not change (so the orbital size and shape remain constant), as well as the inclination i. However, the longitude of the ascending node , the argument of perigee  and the mean anomaly of the epoch M0 experience secular perturbations as follows:

P – period

a – semi-major axis

e – eccentricity

i – inclination

d/dt – time derivatives

The first time derivative means the regression of nodes – the line of apsides rotates in the direction opposite to the orbital motion. There is no regression, if i = 900, i.e. if the orbit is polar. This result is evident due to the symmetry of the problem.

slide11

For example, for the orbit of ISS the calculation gives day  50. If d/dt = 2/365.24, the orbital plane rotates with the same angular velocity as the the Sun. That means that the angle with the solar direction remains unaltered, so that illumination conditions of the surface along the track are the same throughout the year. This is the case of Sun-syncrhonous orbits.

The second time-derivative means rotation of apsides: the orbit rotates in its plane. There is no rotation if cos2i = 1/5, i.e. if i = 63.40 or 116.60. This is the case of Molniya and Tundra orbits: communication satellites at these orbits retain the declination of their perigee unchanged.

The secular perturbations of the second order also do not influence a, e and i, but lead to additional smaller terms in expressions for d/dt, d/dt and dM0/dt. Periodical perturbations introduce a small secular changes to the shape and to the size of the orbit, and they also periodically influence all the elements of the orbit (periods are multiples of P, P/2, P/3 etc.). Non-sphericity of the Earth remains the mayor perturbational factor up to the altitudes of about ~ 50 000 km. At higher altitudes the magnitude of solar and lunar perturbations exceeds them. If the orbital period is a multiple of the Earth’s rotational period, perturbations tend to accumulate due to the resonance. Studies of satellites motion enable to determine with better precision the figure of the Earth and the configuration of its gravity field.

For geostationary satellites an important second-order effect leads to the fact that there are two stable longitudes, that are 750 East and 1050 West, and two unstable longitudes, 1620 East and 110 West. The main reason of this perturbation is that the Earth’s equator is not circular but elliptical (stable points correspond to the minor axis of that ellipse and unstable points to the mayor axis). Objects without station-keeping maneuvers tend to oscillate around one of the first two points, this effect is called libration.

lunar and solar perturbations

Since the Earth has a large natural satellite, the Moon, and the lunar orbit is not very far from the Earth (the semi-major axis of the lunar orbit is about 380 000 km, the eccentricity is ~0.05), gravity of the Moon notably influences the satellites. The Earth also orbits the Sun (the radius of the orbit is about 150 millions of km), which also influences satellites.

The presence of these gravitating objects renders the problem of the satellites motion into a many-body problem of celestial mechanics. This problem cannot be solved analytically. Since the gravity of the Earth influences the satellites motion generally much stronger than that of the Moon and the Sun, this problem may be solved with perturbation theory methods. In addition, since masses of satellites are negligible compared with the masses of the natural celestial bodies, we have a restricted problem (satellites do not influence the motion of the Sun and the planets, but move in their gravity field).

The role of lunisolar perturbation depends on the altitude of the orbit. Below the height of about 20 000 km these perturbations are less than the second-order effects of the Earth’s gravitational anomalies. At the altitude about 50 000 km they become higher than the perturbations caused by the Earth’s oblateness. For most of orbits, lunar perturbations are 2-3 times higher than solar perturbations.

The perturbative acceleration vector of a satellite is the difference between two vectors: the acceleration of the satellite by the perturbing body (the Moon or the Sun) and the acceleration of the central body (the Earth) by the perturbing body.

Lunar and solar perturbations
slide13

Between the bodies it is directed towards the accelerating body, behind the Earth it acts in the opposite direction, since the perturbing body accelerates the Earth stronger than the satellite (thus the perturbing body “repels” the satellite). For the Moon, perturbing accelerations in the general case also has transversal components (perpendicular to the line Earth-Moon). As for the Sun, because of the very high distance its gravity field near the Earth may be treated as nearly parallel, so satellites have perturbing accelerations towards the Sun or back from it.

Effects of lunisolar perturbations strongly depend on the shape of the orbit and on its position relative to the Moon and the Sun. If a perturbing body lies in the orbital plane of a spacecraft, it cannot change this plane; however, due to reciprocal motion of the celestial bodies they leave this plane sooner or later (for instance, if the Sun is in the orbital plane, due to the annual motion it will become perpendicular to it in 3 months).

In general, circular orbits, even with large radii, are stable against lunisolar perturbations, specially if their plane is close to the ecliptic. The main effects are regression of nodes and rotation of apsides. The following approximations may be used.

Daily regression of the ascending node caused by the Moon and the Sun (n – mean motion in revolutions per day).

Daily rotation of apsides caused by the Moon and the Sun.

For highly elliptical orbits, as well as for orbits with semi-major axes of ~ 50 000 km, the role of lunisolar perturbations may be critical. Apogees are influenced in a larger extent:

slide14

first, since they are farther from the Earth and the relative values of perturbing accelerations are higher; and second, since velocities in apogees are smaller, and their relative change is larger. The effect is more pronounced if perturbing accelerations are directed along the velocity vector. An increase of the apogee velocity leads to a rise of the perigee and a drop of the apogee velocity lowers the perigee. The value of change of the perigee height depends mostly on the apogee height and is weakly sensitive to the perigee height. For orbits with apogees of ~ 100 000 km, perigees may change by tens of kilometers per revolution. Constant decrease of the perigee may lead to contact with the atmosphere and re-entry. A resonance with the orbital period of the Moon may appear as well, that may lead to accumulation of effects and a rapid evolution of the orbit. If the apogee is higher than the lunar orbit, the Moon may pull the satellite out from the Earth’s gravity field or the satellite may decay due to lowering of its perigee.

solar radiation pressure

Solar radiation pressure (SRP) is the effect arising due to the fact that radiation exerts a force to the illuminated objects. Satellites are illuminated by the Sun, and the solar light exerts pressure on them. This pressure depends on the reflective properties of the surfaces of the satellite. The radiation pressure may be calculated with the expression

Solar radiation pressure

Thus, for a totally reflective surface the SRP is p  1400/3·108  4.6 ·10-6N/m2.

p – pressure

c – speed of light, 3 ·108m/sec

W – radiation flux (near the Earth ~1400 W/m2, so-called solar constant

– overall surface reflectance (from 0 to 1)

The force which this pressure exerts on a satellite may be found from the following formula

(we neglect the fact that the distance to the Sun changes slightly during the orbital motion)

m – mass of the satellinte

A – area perpendicular to the Sun

pE= W/c  4.6 ·10-6N/m2

The incident radiation is partially absorbed (a), partially reflected specularly (coefficient s) and partially reflected diffusely (d), the sum of these coefficients is unity: a + s+ d = 1. The force related to specularly reflected radiation is directed according to the law of specular reflection, the force related to absorbed radiation is directed along the line “Sun – spacecraft”

slide16

and the force related to diffusely reflected radiation has an intermediate direction. Overall surface reflectance  is a resultant of these factors, in the general case its direction does not coincide with the line “Sun – spacecraft”. The satellite also experiences torque due to its unsymmetrical mass distribution.

Taking into account that the force of SRP depends on the same ballistic coefficient as atmospheric drag, we may easily make sure that effects of SRP by their magnitude are comparable with atmospheric drag at heights of ~ 600-700 km. At lower heights their role is smaller (let us recall that the role of atmospheric drag raises exponentially at lower heights). Like atmospheric drag, smaller objects are more influenced by SRP since mass drops with linear dimensions quicker than the area. In contrast to atmospheric drag, SRP does not vary with altitude. SRP effects are most pronounced for spacecraft with large solar panels, like communication satellites and GPS.

To know effects of SRP, one should account for the shape and reflective properties of the surfaces of the object, the precise Sun location, the attitude of the satellite. Since the satellite may rotate, the effect of SRP is time-varying as reflectivity of its surfaced directed towards the Sun vary. The satellite may get to the Earth’s shadow and be occulted from the solar radiation on certain portions of its orbit (specially on LEO).

As other perturbing forces, SRP exerts the most effect if its vector is coincident with or opposite to the velocity of the spacecraft. Let us examine qualitatively the influence of SRP to the orbit of a GSO satellite if the Sun is in the orbital plane (i.e. during equinox).

slide17

In the upper point 1 of the orbit SRP is opposite to the velocity vector and reduces the velocity, and thus leads to lowering of the perigee in the lower point 2. In the perigee 2 SRP is coincident with the velocity vector and acts to increase it, so the apogee in the point 3 also increases. As a result, the initially circular orbit becomes elliptical. However, in 6 month the direction to the Sun will be opposite, so the effect of SRP will reverse, and the elliptical orbit will transform to circular.

Let us look for some examples. In 1963 a lot of cooper needles were ejected from Midas 6 satellite to form a radio wave reflecting ring along the orbit. Due to SRP, most of the needles re-entered, while the lifetime of Midas 6 is thousands of years (being much larger than the needles, it possesses much higher BC). Inflatable satellites Echo and Pageos, possessing small BC and high reflectance, experienced high influence from SRP. The circular orbit of Echo 1A became elliptical in five months, with the perigee at 900 km and the apogee at 2000 km. In the next half year it became nearly circular again, and the process reiterated. At perigees the satellite was subject to atmospheric drag (because of its low BC) and decayed in 8 years. The upper stage of the launch vehicle which was left at the same initial orbit will stay in space for thousands of years because its high BC.

Use of SPR is the working principle of the so-called solar sail which we shall study later. It can also be used for orbital corrections if the time is not critical. For instance, it was used by mission control to adjust the orbit of the Messenger probe to the Mercury (taking advantage of its proximity to the Sun). This helped to avoid several corrections with the thrusters and thus to safe fuel.

other factors
Other factors
  • The Earth’s albedo: the satellite is illuminated by the Earth also and this radiation also exerts pressure onto its surfaces, “repelling” it out from the Earth. Generally these effects are much smaller than SRP, but in extreme cases (as for sun-synchornous satellites) may average up to ¼ of SRP. Albedo depends on local conditions on the surface and for a dawn/dusk satellite its effective maximum value is ~ 20%.
  • Magnetic effects: the satellite moves through the Earth’s magnetic field and its metallic body and electric circuits interact with this field, which lead to appearance of additional forces.
  • General Relativity effects: for some spacecrafts determination of the orbital position and time is of special importance, in particular, for GPS satellites. So the effects of General Relativity should also be taken into account. The principal effect is apsidal rotation.
  • Residual exhaust: spacecraft may emit residual gases or ions from its surface, that also may slightly perturb their orbital motion.
role of perturbations of different kind

Role of perturbations of different kind

Mass/Area = 200 kg/m2. SRP – solar radiation pressure. Jn – coefficients of the Earth’s oblateness. (Taken from P.W.Fortescue, J.Stark, G.Swinerd, “Spacecraft System Ingeneering”.)

regression of nodes

Regression of nodes

(From the source.)

rotation of the line of apsides

Rotation of the line of apsides

Line of apsides rotates

change of eccentricity due to atmospheric drag

Change of eccentricity due to atmospheric drag

For highly eccentrical orbits, atmospheric drag leads to apogee lowering. For moderately eccentrical orbits, atmospheric drag leads to circularization of the orbit (taken form Graham Swinerd, How Spacecraft fly.)

solar radiation pressure1

Forces due to specular reflection, diffuse reflection and absorption.

The force related to specular reflection is directed along the vector n. The force related to apbsorption is directed along the vector s. Force related to diffuse reflection is directed along the vector s+2n/3. (Taken form the source.)

Solar radiation pressure

SRP increasing eccentricity of a GSO satellite orbit (taken form Graham Swinerd, How Spacecraft fly)