Chapter 5

Orbital Maneuvers

5.1 – Introduction

Orbital maneuvers are carried out to change some of the orbital elements

when the final orbit is achieved via a parking orbit;

to correct injection errors;

to compensate for orbital perturbations (stationkeeping).

The task is usually accomplished by the satellite propulsion system, and

occasionally, in the first two cases, by the rocket last stage. Due to the low

requirements in terms of propulsive effort, the stationkeeping is usually

performed using the thrusters of the attitude control system.

Elementary maneuvers, that involve a maximum of three parameters and

require a maximum of three impulses, are analyzed in the following. When it is

necessary, two or more elementary maneuvers are combined and executed

according to a more complex strategy that permits, in general, a minor propellant

consumption. Nevertheless, it has been demonstrated the any optimal impulsive

maneuver requires a maximum of four burns.

The propellant consumption for the maneuver ( ) is evaluated after the

total velocity change has been computed. The equation presented in Section 4.2

is easily integrated under the hypothesis of constant effective exhaust velocity c

and provides the relationship between the spacecraft final and initial mass, which

is known as rocket equation or Tsiolkovsky’s equation. Therefore

5.1 – One-impulse maneuvers

A single velocity impulse would be sufficient to change all the orbital elements.

Simpler cases are analyzed in this section. Subscripts 1 and 2 denote

characteristics of the initial and final orbit, respectively.

5.1.1 – Adjustment of perigee and apogee height

An efficient way of changing the height of perigee and apogee uses an increment

of velocity provided at the opposite (first) apsis. Misalignment losses and rotation

of the semi-major axis are avoided. Once the required variation of the second

apsis altitude is known, one easily deduces the new length of the semi-major axis

, which is used in the equation energy to compute the velocity at

the first apsis after the burn, which therefore must provide .

For small variations of the second apsis radius, the differential relationship of

section 4.4 can be applied, obtaining

One should note that in some maneuvers the apsides may interchange their role

(from periapsis to apoapsis, and vice versa).

5.1.2 – Simple rotation of the line-of-apsides

A simple rotation of the line of apsides without altering size and shape of the

orbit is obtained by means of one impulsive burn at either point where the initial

and final ellipses intersect on the bisector of the angle . The polar equation of

the trajectory and the constant position of the burn point in a non-rotating frame

imply

which are combined and give

Energy and angular momentum are unchanged by the maneuver; this

corresponds to conserving, respectively, magnitude and tangential component of

the velocity. Therefore, the only permitted change is the sign of the radial

component . The velocity change required to the propulsion system

is the same in either point eligible for the maneuver. The rightmost term has been

obtained using the relationship

5.1.3 – Simple plane change

To change the orientation of the orbit plane requires that the velocity increment

has a component perpendicular to the original plane. A simple plane change

rotates the orbital plane by means of one impulsive burn (r = const), without

altering size and shape of the orbit. Energy and magnitude of the angular

momentum are unchanged, which corresponds to conserving, respectively,

magnitude and tangential component of the velocity. Therefore, the radial

component is also unchanged, while is rotated of the desired angle

From the resulting isosceles triangle in the horizontal plane one obtains

An analysis of the azimuth equation presented in Section 4.7 indicates

that a simple plane change at implies i < (Fig. 5.1); moreover the

inclination after the maneuver cannot be lower than the local latitude ( ). In a

general case the maneuver changes both inclination and longitude of the

ascending node. If the plane change aims to change the inclination, the most

efficient maneuver is carried out when the satellite crosses the equator (therefore

at either node) and is maintained.

0

0.25

0.50

0.75

1.00

-30 30 600

0

10

20

= 30

i (deg)

i/

plane change efficiency(initial inclination 30)

Fig. 5.1 Efficiency of an inclination change carried out at different latitudes

The plane rotation involves a significant velocity change (10% of the

spacecraft velocity for a 5.73 deg rotation) with the associated propellant

expenditure without energy gain. Gravitational losses are not relevant and the

maneuver is better performed where the velocity is low. In most cases, by means

of careful planning, the maneuver can be avoided or executed with a lower cost

in the occasion of a burn aimed to change the spacecraft energy.

5.1.4 – Combined change of apsis altitude and plane orientation

Consider an adjustment of the apsis altitude combined with a plane rotation ,

which is therefore the angle included between the vectors and . Without any

loss of generality, suppose . If the maneuvers are separately performed,

the rotation is conveniently executed before the velocity has been increased, and

the total velocity change is

The velocity increment of the combined maneuver is given by

and the benefit achieved is presented in Fig. 5.2 for different values of

. One should note that, for small angles, , , and the

plane rotation is actually free.

0

0.05

0.10

0.15

0.20

0 30 60 90

vr

0.4

0.2

ve = 0.1

(deg)

vs -

vc

Fig. 5.2 Benefit of the combined change of apsis altitude and plane orientation

5.2 – Two-impulse maneuvers

In this case the maneuver starts at point 1 on the initial orbit, where the

spacecraft is inserted into a transfer orbit (subscript t) that ends at point 2 on the

final orbit.

5.2.1 – Two-impulse rotation of the line-of-apsides

The same maneuver described in section 5.1.2 can be accomplished more

efficiently using two impulsive burns. An analytical solution can be obtained in the

case of small eccentricity e1, under the hypothesis that the angular length of the

transfer is . The initial point of the maneuver has true anomaly on the

transfer orbit, and , on the initial orbit, for the sake of symmetry.

12

/2

1

Fig. 5.3 Two-impulse rotation of the line-of-apsides

The condition provides the semilatus rectum of the transfer ellipse,

when applied to the trajectory equations, and the tangential component of the

velocity at the beginning of the transfer arc

when applied to the angular momentum equations. One should note that does

not depend on et, which can be selected to obtain , thus keeping the

velocity change at the minimum value:

The impulse corresponding to a single burn is

Under the hypothesis of small eccentricity,

and, retaining only the first order terms in e1, the total cost of the two-burn

maneuver

results to be a half of the single-burn maneuver. A further improvement can be

obtained by removing the assumptions, that is, by reducing the transfer angle

and allowing for a small radial component.

5.2.2 – Change of the time of periapsis passage

A change of the time of periapsis passage permits to phase the spacecraft on

its orbit. This maneuver is important for geostationary satellites that need to get

their design station and keep it against the East-West displacement caused by

the Earth’s asphericity. Assuming , the maneuver is accomplished by

moving the satellite on an outer waiting orbit where the spacecraft executes n

complete revolutions. The period of the waiting orbit is selected so different

from the period of the nominal orbit that

or

The rightmost condition, corresponding to an inner orbit, is preferred when is

larger than , and if the perigee of the waiting orbit is high enough above the

Earth atmosphere.

Two equal and opposite are needed: the first puts the spacecraft on

the waiting orbit; the second restores the original trajectory. According to general

considerations, the engine thrust is applied at the perigee and parallel to the

spacecraft velocity. The required is smaller if is reduced by increasing n.

A time constraint is necessary to have a meaningful problem and avoid the

solution with an infinite number of revolution and infinitesimal .

The problem is equivalent to the rendezvous with another spacecraft on

the same orbit. One should note the thrust apparently pushes the chasing

spacecraft away from the chased one.

5.2.3 – Transfer between circular orbits

Consider the transfer of a spacecraft from a circular orbit of radius r1 to another

with radius r2, without reversing the rotation. Without any loss of generality,

assume r2 > r1 (the other case only implies that the velocity-change vectors are in

the opposite direction). The transfer orbit (subscript t) must intersect or at least

be tangent to both the circular orbits

0

0.5

1.0

1.5

2.0

0 1 2 3

ellipses

hyperbolae

H

rA= r2

rP= r1

semi-latus rectum p

ecce

ntrici

ty e

Fig. 5.4 Permissible parameters for the transfer orbit between circular orbits

The permitted values of et and pt are in the shadowed area of Fig. 5.4, inside

which a suitable point is selected. One first computes the energetic parameters

of the transfer orbit

and then velocity and flight path angle soon after the first burn

The first velocity increment is

The second velocity increment at point 2 is evaluated in a similar way.

5.2.4 – Hohmann transfer

The minimum velocity change required for a two-burn transfer between circular

orbits corresponds to using an ellipse, which is tangent to both circles:

On leaving the inner circle, the velocity, parallel to the circular velocity, is

and the velocity increment provided by the first burn is

The velocity on reaching the outer circle

is again parallel, but smaller than the circular velocity. Therefore

The time-of-flight is just half the period of the transfer ellipse

One should note that the Hohmann transfer is the cheapest but the slowest two-

burn transfer between circular orbits. Increasing the apogee of the transfer orbit,

which is kept tangent to the inner circle, soon reduces the time-of-flight.

5.2.5 – Noncoplanar Hohmann transfer

A transfer between two circular inclined orbits is analyzed ( ); a typical

example of application is the geostationary transfer orbit (GTO) that moves a

spacecraft from an inclined LEO to an equatorial GEO. The axis of the Hohmann

ellipse coincides with the intersection of the initial and final orbit planes. Both

impulses provide a combined change of apsis altitude and plane orientation

(Section 5.1.4). The greater part of the plane change is performed at the apogee

of the transfer orbit, where the spacecraft velocity attains the minimum value

during the maneuver. Nevertheless, even for , a small portion of the

rotation (typically 10% for LEO-GEO transfers) can be obtained by the perigee

burn almost without any additional cost.

5.3 – Three-impulse maneuvers

In special circumstances, some maneuvers, which have been analyzed in the

previous sections, are less expensive if executed according to a three-impulse

scheme, which is essentially a combination of two Hohmann transfers; subscript

3 denotes the point where the intermediate impulse is applied.

5.3.1 – Bielliptic transfer

The cost of the Hohmann transfer does not increase continuously, but it reaches

its maximum for (Fig. 5.5). Beyond this value, it is convenient to

begin the mission on a transfer ellipse with apogee at , where the

spacecraft trajectory is not circularized, but a smaller moves the spacecraft

directly into a Hohmann transfer towards the radius , where the spacecraft is

slowed down to make its trajectory circular. The larger is , the smaller is the

total cost of this bielliptic maneuver; one easily realizes that the minimum total

is obtained with a biparabolic transfer. In this case two impulses transfer the

spacecraft from a circular orbit to a minimum energy escape trajectory and vice

versa; a third infinitesimal impulse is given at infinite distance from the main body

to move the spacecraft between two different parabolae.

The biparabolic transfer has better performance than the Hohmann

transfer for . If final radius is between and , also the

bielliptic maneuver may perform better than the Hohmann transfer, but the

intermediate radius should be great enough (Fig. 5.5).

0.45

0.47

0.49

0.51

0.53

0.55

10 20 50 100rB rH 2rH 4rH

bielliptic r3= rHbielliptic r3= 2 rHbielliptic r3= 4 rHbiparabolicHohmann

r2

v/v

1

Fig. 5.5 Comparison of bielliptic, biparabolic and Hohmann transfers ( )

The biparabolic transfer permits a maximum reduction of 8% for , but

the minor propellant consumption of bielliptic and biparabolic transfers is

counterbalanced by the increment of the flight time. A three-impulse maneuver is

rarely used for transfers between coplanar orbits; it becomes more interesting for

noncoplanar transfer as a large part of the plane change can be performed with

the second impulse far away from the central body.

5.3.2 – Three-impulse plane change

A plane change can be obtained using two symmetric Hohmann transfers that

move the spacecraft to and back from a far point where a cheaper rotation is

executed. The rotation is in particular free at infinite distance from the central

body. Assume that the spacecraft is in a low-altitude circular orbit. The velocity

change of a simple plane rotation is equated to the v required to enter and

leave an escape parabola

0

0.2

0.4

0.6

0.8

1.0

0 20 40 600

2

4

6

8

10

biparabolic

simple planechange

v

1

r3

v /

v 1

r 3 / r 1;

1

Fig. 5.6 Three impulse plane rotation for a circular orbit

(dotted lines: rotation at point 3; solid lines: split rotation)

However a bielliptic transfer performs better for a single-burn rotation between

38.94 deg and 60 deg (dotted lines in Fig.5.6). Moreover, a small fraction of

the total rotation can be efficiently obtained (see Section 5.1.4) on leaving the

circular orbit and then again on reentering it. In this case the three-impulse

bielliptic plane change is convenient for any amount of rotation, until the

biparabolic maneuver takes over.

5.3.3 – Three-impulse noncoplanar transfer between circular orbits

Similar concepts also apply to the noncoplanar transfer between circular orbits of

different radii. As it appears in Fig. 5.7, the range of optimality of the bielliptic

transfer becomes narrower as the radius-ratio increases above unity, either if the

rotation is concentrated at the maximum distance , or if it is split among the

three burns.

0

0.2

0.4

0.6

0.8

0 20 40 600

2

4

6

8

10one-rotation Hohmann

21

r3

v

biparabolic

v /

v 1

r 3 / r 1;

1

r2 / r1 = 2

0.4

0.5

0.6

0.7

0.8

0 20 40 600

2

4

6

8

2

1

v

biparabolic

one-rotationHohmann

r3

v /

v 1

r 3 / r 1;

1

r2 / r1 = 6.4

Fig. 5.7 Three impulse transfer between noncoplanar circular orbits

(dotted lines: rotation at point 3; solid lines: split rotation)

In particular, the classical noncoplanar Hohmann transfer is optimal for

deg in the case of the LEO-GEO transfer ( ). For just a little above

this limit, the optimal maneuver is biparabolic.

Chapter 6

Lunar Trajectories

6.1 – The Earth-Moon system

The peculiarity of the lunar trajectories is the relative size of the Earth and Moon,

whose mass ratio is 81.3, which is far larger than any other binary system in the

solar system, the only exception being Pluto and Caron with a mass ratio close to

7. The Earth-Moon average distance, that is, the semi-major axis of the

geocentric lunar orbit, is 384,400 km. The two bodies actually revolve on elliptic

paths about their center of mass, which is distant 4,671 km from the center of the

planet, i.e., about 3/4 of the Earth radius. The Moon's average barycentric orbital

speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds

gives the geocentric lunar average orbital speed, 1.022 km/s.

The computation of a precision lunar trajectory requires the numerical

integration of the equation of motion starting from tentative values for position

and velocity at the injection time, when the spacecraft leaves a LEO parking orbit

to enter the ballistic trajectory aimed at the Moon. Solar perturbations (including

radiation), the oblate shape of the Earth, and mainly the terminal attraction of the

Moon must be taken into account. Because of the complex motion of the Moon,

its position is provided by lunar ephemeris. Approximate analytical methods,

which only take the predominant features of the problem into account, are

required to narrow down the choice of the launch time and injection conditions.

6.2 – Simple Earth-Moon trajectories

A very simple analysis permits to assess the effect of the injection parameters,

namely the radius of the parking orbit r0, the velocity v0, and the flight path angle

0, on the time-of-flight. The analysis assumes that the lunar orbit is circular with

radius R = 384,400 km, and neglects the terminal attraction of the Moon. The

spacecraft trajectory is in the plane of the lunar motion, a condition that actual

trajectories approximately fulfill to avoid expensive plane changes.

One first computes the constant energy and angular momentum

and then the geometric parameters of the ballistic trajectory

2hp

Solving the polar equation of the conic section, one finds the true anomaly at

departure and at the intersection with the lunar orbit (subscript 1)

( )

The time-of-flight is computed using the equations presented in Chapter 2. The

phase angle at departure , i.e., the angle between the probe and the Moon as

seen from the Earth (positive when the spacecraft follows the satellite),

is related to the phase angle at arrival (zero for a direct hit, neglecting the final

attraction of the Moon). Due to the assumption of circularity for both the lunar and

parking orbits, the angle actually fixes the times of the launch opportunities.

The total propulsive effort is evaluated by adding , which is the

theoretical velocity required to attain the parking orbit (Section 4.3), and the

magnitude of the velocity increment on leaving the circular LEO

The results presented in Fig. 6.1 suggest to depart with an impulse parallel to the

circular velocity ( ) from a parking orbit at the minimum altitude that would

permit a sufficient stay, taking into account the decay due to the atmospheric

drag.

25

50

75

100

125

11.2 11.3

z0= 320 km - 0= 0

z0= 220 km - 0= 2z0= 220 km -

0= 0

v = 0

total v (km/s)

time-

of-flig

ht (

hour

)

Fig. 6.1 Approximate time-of-flight of lunar trajectories.

120

140

160

180

10.8 10.9 11.0 11.10.95

1.00

1.05

1.10

1.15

0

1-

0

e

hyperbolaeellipses

injection velocity v0 (km/s)

1- 0 ; 0

(deg

)

ecce

ntric

ity

e

z0= 320 km

Fig. 6.2 Lunar trajectories departing from 320 km circular LEO with 0 = 0.

Other features of the trajectories based on a 320 km altitude LEO are

presented in Fig. 6.2 as a function of the injection velocity v0 ( ). The

minimum injection velocity of 10.82 km/s originates a Hohmann transfer that has

the maximum flight time of about 120 hour. The apogee velocity is 0,188 km/s,

and the velocity relative to the Moon has the opposite direction, resulting in an

impact on the leading edge of the satellite. A modest increment of the injection

velocity significantly reduces the trip time. For the manned Apollo missions, the

life-support requirements led to a flight time of about 72 hour, that also avoided

the unacceptable non-return risk of the hyperbolic trajectories. Further

increments of the injection velocity reduce the flight time and the angle

swept by the lunar probe from the injection point to the lunar intercept. In the

limiting case of infinite injection speed, the trajectory is a straight line with a trip-

time zero, , and impact in the center of the side facing the Earth.

The phase angle at departure presents a stationary point (that is, a

maximum) for = 10.94 km/s. In this condition

and an error on the initial speed, e.g., a higher speed, reduces the swept angle,

but the error is almost exactly compensated ( ) by the less time

required to the Moon to reach the new intersection point. The practical

significance of this condition is extremely scarce, because of the too simplified

model. However, it reminds the reader that actual space missions are designed

looking not only for reduced , but also for safety from errors.

6.3 – The patched-conic approximation

Any prediction of the lunar arrival conditions requires accounting for the terminal

attraction of the Moon. A classical approach is based on the patched-conic

approximation, which is still based on the analytical solution of the two-body

problem. The spacecraft is considered under the only action of the Earth until it

enters the Moon’s sphere of influence: inside it, the Earth attraction is neglected.

The concept of sphere of influence, which was introduced by Laplace, is

conventional, as the transition from geocentric to selenocentric motion is a

gradual process that takes place on a finite arc of the trajectory where both Earth

and Moon affect the spacecraft dynamics equally. Nevertheless this approach is

an acceptable approximation for a preliminary analysis and mainly for evaluating

the injection . The solar perturbation is the main reason that renders the

description of the trajectory after the lunar encounter only qualitative.

The previous assumption of circular motion of the Moon in the same plane of

the spacecraft trajectory is retained. The probe enters, before the apogee, the

lunar sphere of influence whose radius

has been assumed according to Laplace definition.

6.3.1 – Geocentric leg

The geocentric phase of the trajectory may be specified by four initial conditions:

: an iterative process permits the determination of the point where

the spacecraft enters the lunar sphere of influence. The phase angle is

conveniently replaced by another independent variable, that is, the angle

which specifies the position of point 1, where the trajectory crosses the boundary

of the lunar sphere of influence.

The computation of the geocentric leg is carried out using the equations

presented in Section 4.2, with the only remarkable exception of radius and phase

angle at end of the geocentric leg, which are obtained by means of elementary

geometry

The conservation of energy and angular momentum provide the geocentric

velocity and flight path angle at point 1

( , since the arrival occurs prior to apogee). It is worthwhile to note that the

energy at the injection must be sufficient to reach point 1, or, from a

mathematical point of view, to make the argument of the square root positive.

6.3.2 – Selenocentric leg

The spacecraft position and velocity on entering the sphere of influence must be

expressed in a non-rotating selenocentric reference frame (subscript 2) in order

to compute the trajectory around the Moon. The position is simply given by the

radius and the anomaly ; the velocity is

and its direction is described by the angle with the radial direction (a positive

angle means a counterclockwise lunar trajectory). By means of the dot product of

the vector equation with the tangential unit vector , one obtains

The geocentric velocity v1 is quite low (few hundred m/s) and the selenocentric

velocity v2 is mainly due to the Moon velocity. In the most cases v2 is greater than

the lunar escape velocity at the boundary of the sphere of influence and the

spacecraft will approach the Moon along a hyperbola.

The energy and angular momentum of the selenocentric motion

are now computed, and the geometry of the trajectory about the Moon follows

M

Mhp

2

Three possibilities arise:

1. (1738 km), and the spacecraft hits the Moon;

2. Mrr 3 and engines are used to insert the spacecraft into a lunar orbit;

3. and the spacecraft flies by the Moon and crosses again the sphere

of influence.

The first case comprises, together with a destructive impact, a soft landing on the

lunar surface without passing through an intermediate parking orbit. A retrorocket

system or inflatable cushions are required.

Entering in a lunar orbit require a braking maneuver at the periselenium

where the velocity after the maneuver

depends on the semi-major axis ao which is selected for the lunar orbit. The

velocity change is reduced by a low periselenium (minimum gravitational losses)

and a high eccentricity orbit with a far apocenter, which however should permit

the permanent capture by the Moon. Circular orbits are preferred when a

rendezvous is programmed with a vehicle ascending from the lunar surface.

If no action is taken at the periselenium, the spacecraft crosses again the

sphere of influence at point 4 with relative velocity v4 = v2 in the outward direction

( ). The geocentric velocity

is needed for the analysis of the successive geocentric path (subscript 5 denotes

the same exit point on the boundary of the sphere of influence as subscript 4, but

refers to quantities measured in a non-rotating reference frame centered on the

Earth). One should note that the Moon has traveled the angle around

the Earth in the time elapsed during the selenocentric phase; its velocity has

rotated counterclockwise of the same angle. Alternately, the exit point can be

rotated clockwise of the angle , if one is interested in keeping the

horizontal direction of the Earth-Moon line. In this case

(the upper sign for , i.e., counterclockwise lunar trajectory).

A passage in front of the leading edge of the moon rotates clockwise the

relative velocity: one obtains and, by assuming , . On the

contrary, a passage near the trailing edge of the moon rotates counterclockwise

the relative velocity: and the geocentric may be sufficient to escape

from the Earth gravitation. Only the former trajectory is apt to a manned mission

aimed to enter a lunar orbit, as, in the case of failure of the braking maneuver, it

should result into a low-perigee return trajectory.

6.4 – Three-body problem

The restricted three-body problem assumes that the mass of one of the bodies is

negligible and the motion of the two massive bodies is not influenced by the

attraction of the third body. The circular restricted three-body problem is the

special case in which two of the bodies are in circular orbits, an acceptable

approximation for the Sun-Earth and Earth-Moon systems.

In general, the three-body problem cannot be solved analytically (i.e. in

terms of a closed-form solution of known constants and elementary functions),

although approximate solutions can be calculated by numerical methods or

perturbation methods. The three-body problem is however very complex and

difficult to solve and analyze; its solution can be chaotic. The restricted problem

(both circular and elliptical) was worked on extensively by Lagrange in the 18th

century and Poincaré at the end of the 19th century. Poincaré's work on the

restricted three-body problem was the foundation of deterministic chaos theory.

6.4.1 – Jacobi’s integral

The restricted three body problem is usefully analyzed in a reference frame with

its origin in the center of mass of the system. The x-axis coincides with the line

joining the two massive bodies and rotates with angular velocity around the z-

axis. The y-axis completes a right-handed frame. The vector describes the

position of the third body, while and are the positions with respect to the

main bodies of masses M1 and M2, respectively.

The equation of motion of the third body in the rotating frame is

In the circular restricted three-body problem the main bodies are fixed to the

rotating frame and the angular velocity is constant. The previous equation

simplifies to

(hereafter the dot indicates a time-derivative in the rotating frame). The first two

terms on the right-hand side (i.e., the gravitational and centrifugal accelerations)

can be written as the gradient of the potential function

The Coriolis acceleration is a function of the third-body velocity and cannot be

included into a potential function. The equation of motion is written in the form

Scalar multiplication with gives

The potential is not an explicit function of time; therefore

and by substitution and integration one obtains the Jacobi’s integral

where is the velocity of the third body in the rotating frame, and the integration

constant C is known as the Jacobi’s constant. One should note that the half of

the left-hand side of the Jacobi’s integral is the specific mechanical energy of the

third body in its motion relative to the rotating frame. The energy value is

and is kept constant in the circular restricted three-body problem.

6.4.2 – Non-dimensional equations

The equations of the three-body problem are usually made non-dimensional by

using as reference quantities

for masses, distances, and velocities, respectively ( and are the distances

of the primaries from the center of mass). The non-dimensional masses of the

primaries are therefore

and we assume ; the non-dimensional angular velocity is unit, and the

non-dimensional potential is written as

where

as the non-dimensional distances of the greater and smaller bodies from the

origin of the frame are and 1 - , respectively.

6.4.3 – Lagrangian libration points

The circular restricted three-body problem has stationary solutions ( ).

Given two massive bodies in circular orbits around their common center of mass,

there are five positions in space, the Lagrangian libration points, where a third

body, of comparatively negligible mass would maintain its position relative to the

two massive bodies. As seen in the frame which rotates with the same period as

the two co-orbiting bodies, the gravitational fields of two massive bodies

combined with the centrifugal force are in balance at the Lagrangian points.

In the circular problem, there exist five equilibrium points. Three are

collinear with the masses in the rotating frame and are unstable. The remaining

two are located 60 degrees ahead of and behind the smaller mass in its orbit

about the larger mass. For sufficiently small mass ratio of the primaries, these

triangular equilibrium points are stable, such that (nearly) massless particles will

orbit about these points that in turn orbit around the larger primary. Perturbations

may change this scenario, and an object could not remain permanently stable at

any one of these five points. In any case a spacecraft can orbit around them with

modest fuel expenditure to maintain such a position, as the sum of the external

actions is close to zero.

Fig. 6.3 The five Lagrangian points in the three-body system

The L1 point lies on the line defined by the two main bodies, and between

them. An object which orbits the Earth more closely than the Moon would

normally have a shorter orbital period than the Moon, but if the object is directly

between the bodies, then the effect of the lunar gravity is to weaken the force

pulling the object towards the Earth, and therefore increase the orbital period of

the object. The closer to the Moon the object is, the greater this effect is. At the

L1 point, the orbital period of the object becomes exactly equal to the lunar orbital

period.

The L2 point lies on the line defined by the two large masses, beyond the

smaller of the two. On the side of the Moon away from the Earth, the orbital

period of an object would normally be greater than that of the Moon. The extra

pull of the lunar gravity decreases the orbital period of the object, which at the L 2

point has the same period as the Moon.

If the second body has mass M2 much smaller than the mass M1 of the

main body, then L1 and L2 are at approximately equal distances from the

second body (L1 is actually a little closer), given by the radius of the Hill sphere

where R is the distance between the two bodies. The Sun-Earth L1 and L2 are

distant 1,500,000 km from the Earth, and the Earth-Moon points 61,500 km from

the Moon.

The L3 point lies on the line defined by the two large bodies, beyond the

larger of the two. In the Earth-Moon system L3 is on the opposite side of the

Moon, a little further away from the center of mass of the system than the Moon

is, where the combined pull of the Moon and Earth again causes the object to

orbit with the same period as the Moon.

The L4 and L5 points lie at the third point of an equilateral triangle with the

base of the line defined by the two masses, such that the point is respectively

ahead of, or behind, the smaller mass in its orbit around the larger mass. L4 and

L5 are sometimes called triangular Lagrange points or Trojan points.

The gradient vector is zero at the libration points and their exact

position can be found by putting the partial derivatives of the non-dimensional

potential to zero:

The last equation indicates that all the Lagrangian points are in the plane of the

motion of the primaries ( ). The second equation is solved by , which

corresponds to the collinear points, and by , which refers to the

triangular points.

In the former case the first equation becomes a quintic equation in x

with three real solutions. In the latter case

6.4.4 – Surfaces of zero velocity

The potential U is a function only of the position in the rotating frame. The

surfaces in the xyz-space corresponding to a constant value are called

surfaces of zero velocity or Hill surfaces. The constant corresponds to the

total energy of the third body in the rotating frame, and in fact represents the

maximum value of potential energy that the spacecraft can attain by zeroing its

velocity. The spacecraft can only access the region of space with , where

the potential energy is less than its total energy.

The surfaces of zero velocity are symmetrical with respect to the xy-plane;

only their intersection with this plane will be analyzed in the following. Figure 6.x

presents some zero velocity curves and, in particular, those for C1, C2, C3,

corresponding to the values of potential in the collinear Lagrangian points. If is

very large, the zero velocity curves consist of three circles: the largest one has

approximately radius and is centered on the origin of the frame; two smaller

circles enclose the primaries; a spacecraft orbiting the main body with , is

confined to move around it. A spacecraft with can leave the region

around M1 and orbit around M2. For the spacecraft can escape from

the system, but only leaving the region around the primaries from the side of the

second body. This limit disappears for . Any place can be reached when

.

6.4.5 – Lagrangian point stability

The first three Lagrangian points are technically stable only in the plane

perpendicular to the line between the two bodies. This can be seen most easily

by considering the L1 point. If an object located at the L1 point drifted closer to

one of the masses, the gravitational attraction it felt from that mass would be

greater, and it would be pulled closer. However, a test mass displaced

perpendicularly from the central line would feel a force pulling it back towards the

equilibrium point. This is because the lateral components of the two masses'

gravity would add to produce this force, whereas the components along the axis

between them would balance out.

Although the L1, L2, and L3 points are nominally unstable, it turns out that,

at least in the restricted three-body problem, it is possible to find stable periodic

orbits around these points in the plane perpendicular to the line joining the

primaries. These perfectly periodic orbits, referred to as halo orbits, do not exist

in a full n-body dynamical system such as the solar system. However, quasi-

periodic (i.e. bounded but not precisely repeating) Lissajous orbits do exist in the

n-body system. These quasi-periodic orbits are what all libration point missions to

date have used. Although they are not perfectly stable, a relatively modest

propulsive effort can allow a spacecraft to stay in a desired Lissajous orbit for an

extended period of time. It also turns out that, at least in the case of Sun–Earth L 1

missions, it is actually preferable to place the spacecraft in a large amplitude

(100,000–200,000 km) Lissajous orbit instead of having it sit at the libration point,

since this keeps the spacecraft off of the direct Sun–Earth line and thereby

reduces the impacts of solar interference on the Earth–spacecraft

communications links.

The Sun–Earth L1 is ideal for making observations of the Sun. Objects

here are never shadowed by the Earth or the Moon. The sample return capsule

Genesis returned from L1 to Earth in 2004 after collecting solar wind particles

there for three years. The Solar and Heliospheric Observatory (SOHO) is

stationed in a Halo orbit at the L1 and the Advanced Composition Explorer (ACE)

is in a Lissajous orbit, also at the L1 point.

The Sun–Earth L2 offers an exceptionally favorable environment for a

space-based observatory since its instruments can always point away from the

Sun, Earth and Moon while maintaining an unobstructed view to deep space...

The Wilkinson Microwave Anisotropy Probe (WMAP) is already in orbit around

the Sun–Earth L2 and observes the full sky every six months, as the L2 point

follows the Earth around the Sun WMAP. The future Herschel Space

Observatory as well as the proposed James Webb Space Telescope will be

placed at the Sun–Earth L2. Earth–Moon L2 would be a good location for a

communications satellite covering the Moon's far side.

The Sun–Earth L3 is only a place where science fiction stories put a

Counter-Earth planet sharing the same orbit with the Earth but on the opposite

side of the Sun.

By contrast, L4 and L5 are stable equilibrium points, provided the ratio of

the primary masses is larger than 24.96. This is the case for the Sun-Earth and

Earth-Moon systems, though by a small margin in the latter. When a body at

these points is perturbed, it moves away from the point, but the Coriolis force

then acts, and bends the object's path into a stable, kidney bean–shaped orbit

around the point (as seen in the rotating frame of reference).

In the Sun–Jupiter system several thousand asteroids, collectively referred

to as Trojan asteroids, are in orbits around the Sun–Jupiter L4 and L5 points

(Greek and Trojan camp, respectively). Other bodies can be found in the Sun–

Neptune (four bodies around L4) and Sun–Mars (5261 Eureka in L5) systems.

There are no known large bodies in the Sun–Earth system's Trojan points, but

clouds of dust surrounding the L4 and L5 points were discovered in the 1950s.

Clouds of dust, called Kordylewski clouds, may also be present in the L4

and L5 of the Earth–Moon system. There is still controversy as to whether they

actually exist, due to their extreme faintness; they might also be a transient

phenomenon as the L4 and L5 points of the Earth–Moon system are unstable

due to the perturbations of the Sun. Instead, the Saturnian Moon Tethys has two

smaller Moons in its L4 and L5 points, Telesto and Calypso, respectively. The

Saturnian Moon Dione also has two Lagrangian co-orbitals, Helene at its L4 point

and Polydeuces at L5. The Moons wander azimuthally about the Lagrangian

points, with Polydeuces describing the largest deviations, moving up to 32

degrees away from the Saturn–Dione L5 point. Tethys and Dione are hundreds of

times more massive than their "escorts" and Saturn is far more massive still,

which makes the overall system stable.

Appendix – Sphere of influence

Consider a unit-mass spacecraft moving in the proximity of a body of mass M2,

which orbits around a primary body of much greater mass M1. The distance

between the large bodies is R; no forces other than gravitation are considered.

One is interested in defining a region of the space where the action of the second

body is dominant. A precise boundary actually does not exist, but the transition

from the dominance of the primary body to the prevailing action of the second

body is quite smooth. Practical reasons suggest approximating the region with a

sphere, whose radial extension is merely conventional. The most credited

definitions of sphere of influence, which are due to Laplace and Hill, do not

coincide: the former was interested in the transition from a computational model

to another; the latter in the limit altitude of stable orbits.

The concept of sphere of influence is therefore an approximation. Other

forces, such as radiation pressure or the attraction of a fourth body, can be

significant; the third object must also be of small enough mass that it introduces

no additional complications through its own gravity. Orbits just within the sphere

are not stable in the long term; from numerical methods it appears that stable

satellite orbits are inside 1/2 to 1/3 of the Hill radius, with retrograde orbits being

more stable than prograde orbits.

The spheres of influence of the planets in the solar system are given in

Table 6.x. Data refers to the mean distance from the Sun. The planet with the

largest sphere is Neptune; its great distance from the Sun amply compensates

for its small mass relative to Jupiter.

A.6.1 – Laplace sphere of influence

The motion of the spacecraft can be studied in a non-rotating reference frame

with its origin in the center of M2. The only actions on the third body are the

attractions of the central body and of the primary body, which is seen as a

perturbation :

where is the distance of the spacecraft from the second body. If the motion of

the spacecraft is analyzed in a non-rotating frame centered on the primary body,

the main and disturbing forces in the proximity of the second body are instead

where is the distance of the spacecraft from the primary body. According to

Laplace, if one desires to neglect the perturbing action and retain only the main

action, the two approaches can be considered equivalent when

It is convenient to underline this equivalence is conventional, without any

knowledge of the eventual errors.

Table 6.A Spheres of influenceBody /

Laplace Sphere Hill SphereR R R R% (106 km) % (106 km)

Mercury 1,66E-07 0,19 0,112 0,38 0,221Venus 2,45E-06 0,57 0,616 0,93 1,011Earth 3,00E-06 0,62 0,925 1,00 1,497Moon 1,23E-02 17,22 0,066 16,01 0,062Mars 3,23E-07 0,25 0,577 0,48 1,084

Jupiter 9,55E-04 6,19 48,22 6,83 53,15Saturn 2,86E-04 3,82 54,79 4,57 65,45Uranus 4,37E-05 1,80 51,88 2,44 70,24Neptune 5,15E-05 1,93 86,76 2,58 116,18(Pluto) 6,56E-09 0,05 3,15 0,13 7,67

The radius provided by the above equation defines a surface that is

rotationally symmetric about an axis joining the massive bodies. Its shape differs

little from a sphere; the ratio of the largest ( ) and smallest (along the axis)

values of is about 1.15. For convenience the surface is made spherical by

replacing the square root with unity (this is equivalent to select the largest sphere

tangent to the surface). The sphere of radius

is known as Laplace sphere of influence or simply sphere of influence.

The Earth’s sphere of influence has a radius of about 924,000 km, and

comfortably contains the orbit of the Moon, whose sphere of influence extends

out to 66,200 km.

A.6.2 – Hill sphere

The definition of the gravitational sphere of influence, which is known as Hill

sphere, is due to the American astronomer G. W. Hill. It is also called the Roche

sphere because the French astronomer E. Roche independently described it.

The Hill sphere is derived by assuming a reference frame rotating about

the main body with the same angular frequency as the second body, and

considering the three vector fields due to the centrifugal force and the attractions

of the massive bodies. The Hill sphere is the largest sphere within which the sum

of the three fields is directed towards the second body. A small third body can

orbit the second within the Hill sphere, where this resultant force is centripetal.

The Hill sphere extends between the Lagrangian points L1 and L2, which lie along

the line of centers of the two bodies. The region of influence of the second body

is shortest in that direction, and so it acts as the limiting factor for the size of the

Hill sphere. Beyond that distance, a third object in orbit around the second would

spend at least part of its orbit outside the Hill sphere, and would be progressively

perturbed by the main body until would end up orbiting the second body.

The distance of L1 and L2 from the smaller body is obtained by equating

the attractive accelerations of the two primaries to the centrifugal acceleration

where the upper sign applies to L2 and the lower to L1. By replacing

one obtains

and therefore the radius of the Hill sphere of the smaller body

The Hill sphere for the Earth thus extends out to about 1.5 106 km (0.01 AU); the

radius of the Moon’s sphere of influence is close to 61,500 km.

Chapter 7

Interplanetary Trajectories

7.1 – The solar system

The official discover of the heliocentric nature of the solar system was in 1530,

when Copernicus published his revolutionary treatise. Aristarchus of Samos had

already realized that the planets revolve around the Sun, but his belief was

gradually forgot. Nine planets that are usually accompanied by several satellites

encircle the Sun, which has more than 99.8% of the total mass of the system. A

great number of lesser bodies, asteroids or comets, moves in the system, and

additional mass is present as meteors and dust clouds. In comparison with the

Earth, the planets’ total mass is 447 , all the satellites totalize 0.12 , while

the residual mass sums up to only 0.0003 .

The mean distance of the planets from the Sun follows the Bode’s law

after the man who formulated it in 1772. Given the series , the

distance of the planets in Astronomical Units is approximately

with the fifth position occupied by the asteroids and the ninth shared by Neptune

and Pluto, thus suggesting that Pluto might be an escaped satellite of Neptune,

whereas the asteroids might be the remnants of a destroyed body or the matter

destined for a never-born planet. About 220 asteroids are larger than 100 km.

The biggest asteroid is Ceres, which is about 1000 km across. The total mass of

the Asteroid belt is estimated to be 1/35th that of the Moon, and of that total

mass, one-third is accounted for by Ceres alone.

The planets’ orbits are described by five almost constant orbital elements,

whereas the sixth defines the position in the orbit and rapidly changes. Different

sources provide Ephemeris, that are tables of the orbit elements at different

dates for a great number of major and minor bodies. The orbits of the planets are

nearly circular and located in the ecliptic plane, with the exception of the extreme

planets, Mercury (e = 0.206, i = 7.00 deg) and Pluto (e = 0.258, i = 17.14 deg). In

particular Pluto’s perihelion lies inside the orbit of Neptune.

Table 7.1 Orbital Elements of the Planets for the Epoch 01/01/2000 noonPlanet Semi-

major axis Eccentricity Inclination Longitude of ascending node

Longitude of perihelion

Mean longitude at epoch

  a (AU) e i (°) (°) (°) L0 (°)

Mercury 0,3871 0,2056 7,005 48,33 77,46 252,25

Venus 0,7233 0,0068 3,395 76,68 131,60 181,98

Earth 1,0000 0,0167 0,000 0,00 102,94 100,46

Mars 1,5240 0,0934 1,850 49,56 336,06 355,45

Jupiter 5,2030 0,0484 1,304 100,47 14,73 34,40

Saturn 9,5370 0,0539 2,486 113,66 92,60 49,95

Uranus 19,1900 0,0473 0,773 74,02 170,95 313,24

Neptune 30,0700 0,0086 1,770 131,78 44,96 304,88

(Pluto) 39,4800 0,2488 17,140 110,30 224,07 238,93

Some bodies in the solar system are in orbital resonance, a phenomenon

that occurs when two orbiting bodies have periods of revolution that are in a

simple integer ratio, so that they exert a regular gravitational influence on each

other. This can stabilize the orbits and protect them from gravitational

perturbation. For instance, Pluto and some smaller bodies called Plutinos were

saved from ejection by a 3:2 resonance with Neptune. The Trojan asteroids may

be regarded as being protected by a 1:1 resonance with Jupiter. Orbital

resonance can also destabilize one of the orbits. For instance, there is a series of

almost empty lanes in the asteroid belt called Kirkwood gaps where the asteroids

would be in resonance with Jupiter. A Laplace resonance occurs when three or

more orbiting bodies have a simple integer ratio between their orbital periods. For

example, Jupiter's moons Ganymede, Europa, and Io are in a 1:2:4 orbital

resonance.

A companion object of the Earth has been discovered in 1988; the

asteroid 3753 Cruithne is on an elliptical orbit which has almost the same period

of the Earth. Due to close encounters with the planet, this asteroid periodically

alternates between two regular solar orbits. When the asteroid approaches the

Earth (the minimum distance is 12∙106 km), it takes orbital energy from the planet

and moves into a larger and slower orbit. Sometime later, the Earth catches up

with the asteroid and takes the energy back; so the asteroid falls into a smaller,

faster orbit and a new cycle begins. Epimetheus and Janus, satellites of Saturn,

have a similar relationship, though they are of similar masses and so actually

exchange orbits with each other periodically (Janus is roughly 4 times more

massive, but still light enough for its orbit to be altered).

Table 7.2 Physical Characteristics of Sun and PlanetsBody Orbital

periodMean

distance Orbital speed Mass Gravitational

parameter Equatorial

radius Inclination of

equator to orbit

  (years) (106 km) (km/s) (Earth=1) (km3/s2) (km) (°)

Sun - - - 333432 1,327E+11 695000 7,25

Mercury 0,241 57,9 47,87 0,055 2,203E+04 2440 2,11

Venus 0,615 108,2 35,02 0,815 3,249E+05 6052 177,40

Earth 1,000 149,6 29,78 1,000 3,986E+05 6378 23,45

Mars 1,881 227,9 24,08 0,107 4,283E+04 3396 23,98

Jupiter 11,86 778,5 13,07 318,0 1,267E+08 71492 3,08

Saturn 29,66 1433,0 9,69 95,2 3,793E+07 60268 26,73

Uranus 84,32 2877,0 6,81 14,6 5,794E+06 25559 97,92

Neptune 164,8 4503,0 5,43 17,3 6,837E+06 24764 28,80

(Pluto) 248,1 5906,0 4,67 0,0021 8,710E+02 1151 119,59

Comets are small bodies in the solar system characterized by a nucleus,

generally less than 50km across, which is composed of rock, dust, and ice. When

a comet approaches the inner solar system, radiation from the Sun causes its

outer layers of ice to evaporate. The streams of released dust and gas form a

huge but extremely tenuous coma, which can extend over 150 million km (1 AU).

The coma become visible from the Earth when a comet passes closes to the

Sun, the dust reflecting sunlight directly and the gases glowing due to ionization.

The force exerted on the coma by the solar wind and radiation pressure cause

two distinct tails to form pointing away from the Sun in slightly different directions.

The dust is left behind in the comet's orbit so that it often forms a curved tail; the

ionized-gas tail always points directly away from the Sun, since the gas is more

affected by the solar wind than dust is, and follows magnetic field lines rather

than an orbital trajectory.

Fig. 7.1 A comet high-ellipticity orbit (note the two distinct tails)

Comets are classified according to their orbital periods. Short period

comets have orbits of less than 200 years (comet Encke never is farther from the

Sun than Jupiter). Long period comets have larger orbits but remain

gravitationally bound to the Sun. Single-apparition comets have parabolic or

hyperbolic orbits which will cause them to permanently exit the solar system after

one pass by the Sun. Short-period comets are thought to originate in the Kuiper

belt. which is an area of the solar system extending from within the orbit of

Neptune (at 30 AU) to 50 AU from the Sun, at inclinations consistent with the

ecliptic. Long-period comets are believed to originate in the 50,000 AU distant

Oort cloud, which is a spherical cloud of debris left over from the condensation of

the solar nebula and containing a large amount of water in a solid state. Some of

these objects were perturbed by gravitational interactions and fell from their

circular orbits into extremely elliptical orbits that bring them very close to the Sun.

Because of their low masses, and their elliptical orbits which frequently take them

close to the giant planets, cometary orbits are often perturbed and constantly

evolving. Some are moved into sungrazing orbits that destroy the comets when

they approach the Sun, while others are thrown out of the solar system forever.

7.2 – Heliocentric transfer

The computation of a precision trajectory for an interplanetary mission requires

the numerical integration of the complete equation of motion where all

perturbation effects are taken into account. The initial values of the state

variables, namely position and velocity, are modified until a satisfying mission is

found.

The spacecraft spends most of the trip-time under the dominant

gravitational attraction of the Sun and the perturbations caused by the planets

are negligible; only for brief periods the trajectory is shaped by the departure and

arrival planets. In these circumstances the patched-conic model is absolutely

adequate, and this simpler approach is generally adopted for preliminary

analyses. In this case the study of an interplanetary mission begins with the

heliocentric transfer orbit.

The Sun attraction is neglected during the planetocentric legs, and, for the

sake of simplicity, the velocity relative to the planet, on exiting or entering the

sphere of influence, is assumed equal to the hyperbolic excess velocity. The

dimension of the sphere can be neglected in comparison with the distance from

the Sun, and the extremes of the heliocentric leg are assumed to coincide with

the actual positions of the planets.

The heliocentric velocity at departure

is the sum of the Earth velocity and the hyperbolic excess velocity, i.e., the

spacecraft velocity relative to the Earth. As , due to the modest

capabilities of present space propulsion, the maximum angle between and

is quite small (Fig. 7.2). In particular the heliocentric leg can be in a plane that

has only a modest inclination away from the ecliptic plane. In any case the Earth

velocity is better exploited is the spacecraft departs either in the same or in the

opposite direction.

7.2 1– Ideal planets’ orbits

The orbit of most of the planets can be considered circular and coplanar, and the

most efficient transfer between them uses the Hohmann ellipse. One easily

computes the heliocentric velocity and then the hyperbolic excess velocity on

leaving the Earth sphere of influence

A transfer to the inner planets requires that the spacecraft is launched in the

direction opposite to the Earth’s orbital motion. A hyperbolic excess velocity close

to the Earth velocity would be necessary to hit the Sun. A greater propulsive

effort would be necessary to insert a spacecraft into a retrograde heliocentric

orbit directly from the Earth.

The mission could be carried out using any of the ellipses (present

technology does not permit hyperbolae) that intersect or are tangent to the

planets’ circular orbits. In the most general case, an ellipse crosses each circle in

two points providing four different missions, if one neglect the option of

performing one or more complete revolutions around the Sun.

A transfer other than the Hohmann ellipse (Fig. 7.3) may be preferred if

presents minor sensitivity to injection errors, permits an Earth return trajectory, or

simply reduces the trip time. Power requirements and solar interference on

communications between ground stations and spacecraft depend on the distance

and phase angle between the planets at the arrival time

and influence the selection of the heliocentric transfer orbit.

Departure occurs when the phase angle between the planets is

If departure must be delayed, the same mission can be flown after a synodic

period , corresponding to time needed to the faster planet to perform exactly

one more revolution around the sun than the other planet

with as greater as closer the planets’ orbits. Therefore, the longest synodic

periods with the Earth pertain to Venus (1.6 years) and Mars (2.13 years).

A more expensive departure with parallel to , but larger than in the

Hohmann transfer, permits a minor trip time; on the other hand, the hyperbolic

excess velocity at arrival increases, together with the propellant consumption if a

braking maneuver is planned. This kind of trajectory is considered when some

payload is traded off for opening a launch window: using all the allocated

propellant, the target planet (Mars in Fig. 7.4) can be intercepted either at or

. In the same figure the Mars position at departure is brought back in , as

between and the planet moves faster than the spacecraft. Obviously the

mission can be successfully accomplished departing with less , i.e. using less

than the allocated propellant, when Mars is between and and reaching the

planet in a point of its circular orbit between and . Figure 7.4 is drawn by

artificially keeping in 1 the Earth at departure; the phase angle between the

planets decreases when time elapses, and Mars moves from , where the

launch window opens, to , where it is definitely closed.

7.2.3 – Accounting for real planets’ orbits

The Hohmann analysis neglects eccentricity and inclination of the planets’ orbits.

More realistic results are provided by a two-parameter analysis which searches

for the best departure and arrival times. When one assumes suitable values of

these parameters, the trip time is fixed and ephemeris furnish the position of the

start and end points. If one excludes midcourse maneuvers, the interplanetary

trajectory is in the plane containing Sun, Earth at departure time, and target

planet at arrival time. The solution of the Lambert problem provides the orbital

parameters of the interplanetary conic leg and the hyperbolic excess velocities at

departure and arrival.

Two suitable trajectories are usually found for transfer angles either

less or greater than . In fact, when the target planet, which is out of the ecliptic

plane, is intercepted in opposition with the Earth initial position, the plane of the

Lambert ballistic trajectory has often a too high inclination. In such a case, a

different strategy may be useful. The mission is first designed in the ecliptic

plane, i.e., neglecting the inclination of the target. The spacecraft departs and

moves in the ecliptic plane until reaches the quadrature with the arrival point. A

plane change is performed there (90° before arrival) with a midcourse impulse

being the declination of the planet at the end time. This maneuver keeps the

inclination of the second part of the trajectory at the minimum value.

Fig. 7.5 ?

7.3 – Earth departure trajectory

The initial branch of the trajectory is a hyperbola inside the Earth’s sphere

of influence. One should note that a single impulse at ground level provides both

the evasion from the Earth and the injection into the interplanetary leg. This

would correspond to the most efficient mission, as gravitational losses are

minimized.

Nevertheless departure from a parking LEO is usually preferred. Splitting

the maneuver permits post launch checks and some adjustment of the planned

mission, if required. Precision is also improved by the reduced amount of the final

burn and corresponding velocity increment.

7.3.1 – Departure from ground

The spacecraft velocity at departure from the ground is obtained by

considering the conservation of energy during the geocentric leg

The rightmost equation highlights the leveraging effect of using the thrust in the

close proximity of the Earth. On converse, the requirements in terms of accuracy

in the burn-off timing, and therefore in the injection velocity, are quite demanding.

The v requirements for the simple flyby (that is, a close approach) of the

solar system planets is presented in Table 7.3, under the assumption of circular

and coplanar orbits. Venus (11.45 km/s) and Mars (11.55 km/s) require an effort

which barely rises above the second cosmic velocity vII. The results are

presented in Fig. 7.6, and compared to the total velocity increment

which is required by the wrong strategy that would apply a second impulse after

a parabolic evasion from the Earth attraction.

Table 7.3 Missions to solar system planets    Dv departing from Earth surface

arrival synodic simple capture intoplanet period flyby ellipse circle

  years km/s km/s km/sMercury 0.317 13.47 19.92 21.03Venus 1.599 11.45 11.92 14.70Mars 2.135 11.55 12.27 13.64Jupiter 1.092 14.22 14.63 31.19Saturn 1.035 15.19 15.68 25.55Uranus 1.012 15.87 16.41 22.38Neptune 1.006 16.14 16.52 23.04Pluto 1.004 16.26 17.47 18.94

The escape hyperbola, with semiaxis , is shown in Fig. 7.7.

After the angular position of the departure point has been selected, one obtains

e, , and , by solving the system

The direction of the launch velocity is given by

The launch from points with is not permitted; one has to wait that the Earth

rotation moves the launch base outside of the forbidden zone.

0

4

8

12

8 12 16 20 24

v0

vw

Escape fromsolar system

NeptunePluto

UranusSaturn

Jupiter

Mercury

VenusMars

Departure v (km/s)

Hyp

erbo

lic e

xces

s ve

loci

ty v

(km

/s)

Fig. 7.6 v requirements for the simple flyby of the solar system planets.

7.3.2 – Departure from LEO

The same equations of the previous section apply, but is the

radius of parking LEO. Its altitude derives from a compromise between the

gravitational losses and the atmospheric drag that permits the completion of a

sufficient number of orbits. Misalignment losses can be avoided by enforcing

(Fig. 7.8)

and the departure impulse is

The minor semiaxis of the hyperbola, that is the dislocation of the

asymptote , is negligible in comparison with interplanetary distances:

in fact, only the direction is strictly prescribed and the nominal exits from the

center of the Earth and pierces the surface at point P, whose position on the

terrestrial surface is described by declination P and right ascension P. It is

convenient that ascent trajectory, parking orbit, and hyperbola are kept coplanar:

any plane containing the nominal can be selected, and the spacecraft can be

inserted in LEO from any place on the Earth surface.

0

30

60

90

0 30 60 90 120 150 180

L

maximum payload

a

b

sinL > cosP / cosL

L > 118°L < 35°

launch azimuth L (°)

poin

t P d

eclin

atio

n

P

(°)

Fig. 7.9 ingrandire tic

The whole maneuver is in a plane that, at lift-off, contains point P

and launch base L. The inclination is related to the launch azimuth L according

to

which implies the constraint (eastward launch is assumed)

which is only effective when .

-90

-45

45

90

0

L

-180 -90 0 90 180

Pa

Pb

Lb

La2La1

right ascension (°)

decli

natio

n

(°)

Fig. 7.10

The launch corridor is a further specific constraint of the base.

Figure 7.9 refers to Cape Kennnedy ( ) where the permissible azimuth

range is . Propulsive requirements demand the launch azimuth

closest to 90°, which would provide the maximum payload. Once the launch

azimuth has been selected, the inclination is fixed, but corresponds to two planes

with different P. Two daily opportunities (Fig. 7.10) arise as the rocket should be

launched, in theory, when the base crosses either plane; a launch window of

about one hour is considered necessary in the practice.

7.4 – Arrival to the target planet

The spacecraft approaches with relative velocity the target planet, which is

often the final destination of the mission; in these cases a landing or a permanent

capture is planned. Many devices can be used for the braking maneuver: only

chemical propulsion, that implies impulsive maneuvers, will be considered in the

following. In other cases only a close approach is sought, usually with the aim of

receiving a gravity assist from the planet.

The arrival is analyzed using a right-handed planetocentric

reference frame based on a plane normal to the hyperbolic excess velocity. The

unit vector has the same direction as ; is parallel to the ecliptic plane,

points towards the celestial south. In this frame the trajectory is a hyperbola;

vector connects the center of the planet to the aiming point where the incoming

asymptote of the hyperbola crosses the fundamental plane.

The relative motion of the vehicle is in the plane containing and

. Energy and angular momentum in the planetocentric frame are written using

radius and velocity at pericenter (subscript 3)

and combined in order to remove v3, getting a second order equation for r3

which provides the minimum distance from the planet

The rotation of the relative velocity is a function of the

eccentricity

A large rotation (small ) is obtained when the planet is massive (large ) and

the spacecraft has a close approach with low relative velocity.

The plane of the planetocentric trajectory and the pericenter radius

are adjusted by tuning the vector before reaching the sphere of influence, for

instance with a midcourse impulse. In the simple case of circular and coplanar

planets’ orbits, the desired B can be obtained by delaying or advancing the

departure from the Earth. After computing in the heliocentric frame the velocity V2

and its angle 2 above the horizon, the same quantities in the planetocentric

frame are obtained using

The phase angle at departure

is purposely reduced (increased) for a passage in front of (behind) an exterior

planet.

The magnitude of must be larger than the capture radius to

avoid an impact on the surface or an aerodynamic brake. To this end, the

minimum value of the pericenter is the radius of the planet augmented by the

width of the atmosphere

The capture radius is the minimum pericenter times a function (greater than

unity) of the hyperbolic excess velocity and the escape velocity at .

7.4.1 – Capture into circular orbit

A single engine burn at the hyperbola pericenter would be sufficient for the

capture into a circular orbit with radius . The required velocity change is

A simple derivative provides the value that corresponds to the best

compromise between the conflicting requirements of containing energy reduction

and gravitational losses.

A two-impulse maneuver allows a minimum-altitude braking which

inserts the spacecraft into a Hohmann ellipse with . At the

following apocenter the energy is partially restored to circularize the orbit. The

contradictory use of the propellant reduces the efficiency of the maneuver and

the resulting

is less than only for .

0

0.1

0.2

0.3

0.4

0.5

1 10 100 1000

vI

vII

vIII

circular orbit radius

capt

ure v

Fig. 7.11 Comparison of braking maneuvers for circular capture (vinf = ?)

A more efficient bielliptic maneuver is permitted by the addition of a

third impulse. In the limit case of biparabolic transfer the propellant is only used

to reduce the spacecraft energy: an intense braking at the minimum radius is

followed by a second one that reduces again the energy and circularizes the

orbit. The total cost

diminishes by increasing the final radius (i.e., the planetocentric energy).

The braking maneuvers are compared in Fig. 7.7 where radii and

velocities are made non-dimensional using the minimum radius and the

corresponding circular velocity as reference values.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4

parabola

ellipse (rmax= 50 rmin)

capture v

hype

rbol

ic e

xces

s ve

loci

ty

v

Fig. 7.12 Capture into parabolic or highly elliptic planetocentric orbit

7.4.2 – Capture into elliptic orbit

A braking maneuver that inserts the spacecraft into a high-eccentricity high-

energy elliptic orbit combines a low energy reduction with a very efficient use of

propulsion at the minimum permissible distance from the planet. The limit case is

the quasi-capture into a parabolic trajectory; the three-body problem provides a

more realistic guess at the minimum that guarantees the capture. Figure 7.12

presents the required non-dimensional for the parabolic limit case and for the

more conservative case of permanent capture into an elliptic orbit with

.

7.4.3 – Gravity assist

A planet is quite often approached during an interplanetary mission to obtain a

gravity assist. An extreme form of the maneuver if first presented. A Hohmann

transfer is used to sent a spacecraft towards a point-mass outer planet moving

with heliocentric velocity . At aphelion the spacecraft, which has the velocity

in the "fixed" solar frame, is approaching head-on the planet at the relative speed

. A loop around and behind the point-mass planet in an extremely

eccentric hyperbolic orbit provides a virtual 180-degree turn, as illustrated in Fig.

7.f.

0

5

10

15

20

25

30

35

Mercury Venus Mars Jupiter Saturn Uranus Neptune Pluto

Del

taV

(km

/s)

FlybyO. ell.O. circ.

Fig. 7.13 requirements for simple flyby and capture into different orbits

From the planet's perspective the spacecraft will also recede at the speed

relative to the planet, but the planet is still moving at (virtually) the speed , so

the spacecraft heliocentric velocity will be eventually . The

net effect is almost as if the spacecraft "bounced" off the front of the planet.

U=Vt ; V2=-v

Using a rigorous approach, conservation of kinetic energy and momentum

before and after the interaction requires

that are posed in the form ( )

whose ratio

is combined with the moment equation to eliminate either or , giving

Since q is virtually zero (the probe has negligible mass compared with the planet)

confirming previous intuitive result.

The maneuver is more complex in practical cases; the rotation of

the relative velocity is lower than 180° and is not in the hyperbola plane. By

neglecting any change of the planet’s velocity

and assuming , as the radius of the sphere of influence is small with

respect to the distance from the Sun, the change of the heliocentric energy is

The vector , with magnitude , is parallel to the hyperbola axis; by

considering that ,

where is the angle between and the component of the planet velocity

in the hyperbola plane. The largest energy change

is obtained when the hyperbola axis is parallel to the planet velocity. The flyby

effectiveness is enhanced by a close approach to a planet with large mass

and/or velocity; for different reasons, Jupiter and Venus flybys are frequently

exploited. An optimal value of the approach velocity exists, as the energy change

is zero for either zero or infinite ( , in the latter case).

0

45

90

135

180

0 5 10 15 20

Mars

Venus

Earth

Uranus

Saturn

Jupiter

hyperbolic excess velocity v (km/s)

defle

ctio

n an

gle

-

2

(°)

0

100

200

300

400

0 5 10 15 20

Earth

Venus

Mars

Uranus

Saturn

Jupiter

hyperbolic excess velocity v (km/s)

ener

gy in

crea

se

(km

2 /s2 )

Fig. 7.14 Deflection angle and energy increase assuming hyperbola axis parallel

to the planet velocity and pericenter at surface radius times 1.1

7.5 – Tisserand's Criterion

The orbital parameters of the heliocentric trajectories before and after the flyby

must satisfy a relation that the astronomer Francois Felix Tisserand introduced

as an useful criterion to determine whether or not an observed orbiting body,

such as a comet or an asteroid, is the same as a previously observed orbiting

body. While all the orbital parameters of an object orbiting the Sun during the

close encounter with another massive body (e.g. Jupiter) can be changed

dramatically, the value of a function of these parameters, called Tisserand's

parameter, is approximately conserved.

Tisserand's parameter is derived in the circular restricted three-body

system under the assumption that one of the two primaries is much smaller than

the other. These conditions are satisfied for example for the Sun-Jupiter system

with a comet or a spacecraft being the third mass. The Jacobi integral, which is

the constant of motion through the encounter, is expressed using the orbital

parameters of the heliocentric trajectory, which remain constant in two-body

problem, i.e., when the smallest mass is far from the second body.

Under the assumption , the barycenter of the three-body system

approximately coincides with the Sun’s center of mass ( ). The velocity

in the rotating frame of the three-body problem is related to the velocity in a

heliocentric non-rotating frame using

and the Jacobi’s integral can also be expressed in terms of absolute velocity as

Outside the sphere of influence of the second body, the potential energy related

to its attraction can be neglected, and one obtains

In an inertial reference frame, a combination of the total specific energy and

angular momentum of the third body, which are both constant in the two-body

problem approximation, remains constant through the flyby.

Observing that

one easily obtains Tisserand’s relation

where is the angle between and , that is, the inclination of the spacecraft

orbital plane with respect to the planet’s heliocentric orbit; is the semi-major

axis of the heliocentric orbit, made non-dimensional using the distance of the

second body from the Sun.

If Tisserand's parameters, computed for a specific planet and two bodies

observed on different orbits, are nearly the same, one may conclude, according

to Tisserand’s Criterion, that the observations are of the same body, which has

encountered the planet between the observations.

7.A – Appendix: Dwarf Planets

The word planet comes from the Greek word πλανήτης (wanderer) meaning that

planets were originally defined as objects that moved in the night sky with

respect to the background of fixed stars. The five planets closest to Earth can be

discerned with the naked eye without much difficulty and were known to the

ancients prior to the invention of the telescope. Mercury and Venus are only

visible in twilight hours as their orbits are interior to the Earth's orbit. Venus is the

most prominent planet, being the third brightest object in the sky after the Sun

and the Moon. Mercury is more difficult to see due to its proximity to the Sun.

Mars is at its brightest when it is in opposition to the Earth, which occurs

approximately every two years. Jupiter and Saturn are the largest of the five

planets, but are farther from the sun, and therefore receive less sunlight.

Nonetheless, Jupiter is often the next brightest object in the sky after Venus.

Saturn's luminosity is often enhanced by its rings, which reflect light back toward

the Earth to varying degrees, depending on their inclination to the ecliptic.

Uranus is visible to the naked eye in principle on very clear nights, but its

discovery on March 13, 1781, was made using a telescope. Neptune was the first

planet found by mathematical prediction based on unexpected changes in the

orbit of Uranus that led astronomers to deduce that its orbit was subject to

gravitational perturbation by an unknown planet. Neptune was subsequently

found on September 23, 1846, within a degree of its predicted position.

Pluto was discovered in 1930 at the Lowell Observatory by an American

astronomer, Clyde Tombaugh, who was continuing the search for an elusive

planet that Lowell had believed (incorrectly) to be responsible for perturbing the

orbits of Uranus and Neptune. Pluto, once known as the smallest, coldest, and

most distant planet, takes 248 years to orbit the Sun. Between 1979 and 1999,

Pluto's highly elliptical orbit brought it closer to the Sun than Neptune. Because

Pluto is so small and far away, it is difficult to observe from Earth. Most of what

we know about Pluto we have learned from the Hubble Space Telescope.

Pluto is about two-thirds the diameter of Earth's Moon and may have a

rocky core surrounded by a mantle of water ice. Due to its lower density, its mass

is about one-sixth that of the Moon. Pluto appears to have a bright layer of frozen

methane, nitrogen, and carbon monoxide on its surface. While it is close to the

Sun, these ices thaw, rise, and temporarily form a thin atmosphere, with a

pressure one one-millionth that of Earth's atmosphere. Pluto's low gravity (about

6% of Earth's) causes the atmosphere to be much more extended in altitude than

our planet's. Because Pluto's orbit is so elliptical, Pluto grows much colder during

the part of each orbit when it is traveling away from the Sun. During this time, the

bulk of the planet's atmosphere freezes.

In 1978, astronomers discovered that Pluto has a satellite (moon), which

they named Charon. In Greek mythology, Charon was the boatman who carried

the souls of the dead to the underworld kingdom that in Roman mythology was

ruled by the god Pluto. At about 1,186 km, Charon's diameter is a little more than

half of Pluto's, whereas the mass is about one sixth. Pluto and Charon are thus

essentially a double planet. Charon's surface is covered with dirty water ice and

doesn't reflect as much light as Pluto's surface. One theory is that the materials

that formed Charon were blasted out of Pluto in a collision. That's very similar to

the way in which our own moon is thought to have been created. The duo's

gravity has locked them into a mutually synchronous orbit, which keeps each one

facing the other with the same side. Many moons - including our own - keep the

same hemisphere facing their planet. But this is the only case in which the planet

always presents the same hemisphere to its moon. If you stood on one and

watched the other, it would appear to hover in place, never moving across the

sky. Two additional moons Nix and Hydra, were discovered using the Hubble

Space Telescope in 2005.

For almost 50 years Pluto was thought to be larger than Mercury, but with

the discovery in 1978 of Charon, it became possible to measure the mass of

Pluto accurately and it was noticed that actual mass was much smaller than the

initial estimates. It was roughly one-twentieth the mass of Mercury, which made

Pluto by far the smallest planet. Although it was still more than ten times as

massive as the largest object in the asteroid belt, Ceres, it was one-fifth that of

Earth's Moon. Furthermore, having some unusual characteristics such as large

orbital eccentricity and a high orbital inclination, it became evident it was a

completely different kind of body from any of the other planets.

In the 1990s, astronomers began to find objects in the same region of

space as Pluto (now known as the Kuiper belt), and some even farther away.

Many of these shared some of the key orbital characteristics of Pluto, and Pluto

started being seen as the largest member of a population of icy bodies located in

a region of the solar system beyond the orbit of Neptune and often simply labeled

as Trans-Neptunian Objects (TNOs). This led some astronomers to stop referring

to Pluto as a planet. By 2005, three other bodies comparable to Pluto in terms of

size and orbit (Quaoar, Sedna, and Eris) had been reported in the scientific

literature. It became clear that either they would also have to be classified as

planets, or Pluto would have to be reclassified. Astronomers were also confident

that more objects as large as Pluto would be discovered, and the number of

planets would start growing quickly if Pluto were to remain a planet.

In 2006, Eris (then known as 2003 UB313) was determined to be slightly

more massive than Pluto and that it too had a satellite; some reports unofficially

referred to it as the tenth planet. A better definition of the term planet became

necessary and on August 24, 2006, the International Astronomical Union (IAU)

resolved “that planets and other bodies, except satellites, in our Solar System be

defined into three distinct categories in the following way:

A planet is a celestial body that (a) is in orbit around the Sun, (b) has

sufficient mass for its self-gravity to overcome rigid body forces so that it

assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared

the neighborhood around its orbit.

A dwarf planet is a celestial body that (a) is in orbit around the Sun, (b) has

sufficient mass for its self-gravity to overcome rigid body forces so that it

assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared

the neighborhood around its orbit, and (d) is not a satellite.

All other objects, except satellites, orbiting the Sun shall be referred to

collectively as small Solar System bodies.”

This classification states that a planet is required to be massive enough to

have cleared the neighborhood of its orbit by capturing or ejecting into another

orbit any object that ventures nearby. This measure knocks out Pluto and

2003UB313 (Eris), which orbit among the icy wrecks of the Kuiper Belt, and

Ceres, which is in the asteroid belt. The IAU Resolution means that the Solar

System officially consists of eight planets: Mercury, Venus, Earth, Mars, Jupiter,

Saturn, Uranus and Neptune (mnemonic: My Very Excellent Mother Just Served

Us Nachos, which replaces My Very Excellent Mother Just Served Us Nine

Pizzas).

All objects that orbit the Sun and are not massive enough to be rounded

by their own gravity are small bodies. This class currently includes most of the

Solar System asteroids, near-Earth objects (NEOs), Mars and Jupiter Trojan

asteroids, most Centaurs, most Trans-Neptunian Objects (TNOs) and comets.

The intermediate category comprises the dwarf planets, that are generally

smaller than Mercury and orbit in zones containing many other objects, for

example, within the asteroid belt. They need sufficient mass to overcome rigid

body forces and achieve hydrostatic equilibrium, meaning that a layer of liquid

placed on their surface would keep the same shape, apart from small-scale

surface features such as craters and fissures. This does not mean that the body

(as well as the liquid surface) is a sphere; the faster a body rotates, the more

oblate or even scalene it becomes. The extreme example of a non-spherical

body in hydrostatic equilibrium is Haumea, which is twice as long along its major

axis as it is at the poles.

While there are no specific limits for the mass of a planet or dwarf planet,

empirical observations suggest that the lower bound may vary according to the

composition of the object. For example, in the asteroid belt, Ceres, with a

diameter of 975 km, is the only object known to presently be self-rounded

(though Vesta may once have been). Therefore, the limit where other rocky-ice

bodies like Ceres become rounded might be somewhere around 900 km. Icy

bodies like TNOs have less rigid interiors and therefore more easily relax under

their self-gravity into a rounded shape. The smallest icy body known to have

achieved hydrostatic equilibrium is Mimas, while the largest irregular one is

Proteus; both average slightly more than 400 km in diameter, which should be

the lower limit for an icy body to achieve roundness.

Another 2006 IAU's Resolution recognized Pluto as "the prototype of a

new category of trans-Neptunian objects". Almost two years later the IAU has

defined plutoids all trans-neptunian dwarf planets similar to Pluto. While all

plutoids are dwarf planets, not all dwarf planets are plutoids, as is the case with

Ceres. Satellites of plutoids are not plutoids themselves, even if they are massive

enough that their shape is dictated by self-gravity.

As of 2008, the IAU has classified five celestial bodies as dwarf planets.

Two of these, Ceres and Pluto, have been observed in enough detail to

demonstrate that they fit the definition. Eris has been accepted as a dwarf planet

because it is more massive than Pluto. The IAU decided that TNOs with an

absolute magnitude less than +1 (and hence a minimum diameter of 838 km)

have to be considered dwarf planets. The only two such objects known at the

time, Makemake and Haumea, were declared to be dwarf planets.

Ceres: discovered on January 1, 1801, 45 years before Neptune, by Giuseppe

Piazzi, Ceres has a diameter of about 950 kilometers and is by far the largest

and most massive known body in the asteroid belt, as it contains approximately a

third of the belt's total mass. It was considered a planet for half a century, before

numerous other bodies were discovered in the same region and Ceres lost its

planetary status. For more than a century, Ceres has been referred to as an

asteroid or minor planet. Classified as a dwarf planet on September 13, 2006,

because it is massive enough to have self-gravity pulling itself into a nearly round

shape. Ceres orbits within the asteroid belt where many other asteroids come

close to the orbital path of Ceres.

Pluto: discovered on February 18, 1930, classified as a planet until August

24, 2006, when was numbered 134340 and reclassified as a dwarf planet.

Pluto is the second largest object in the group of plutoids.

Eris: discovered on January 5, 2005, provisionally named 2003 UB313, or

Xena, is now called Eris, after the Greek goddess of discord and strife, a

name which the discoverer found fitting in the light of the academic turmoil

that followed its discovery. Eris’s moon is Dysnomia, the demon goddess of

lawlessness and the daughter of Eris. Called the tenth planet in media

reports, Eris has a diameter of 3,000 km, which is 700 km larger than Pluto.

Eris is significant because it is now known as the largest dwarf planet and

more distinctly, a plutoid orbiting in the scattered disk, a region in the Solar

System even more distant than the Kuiper Belt. It is the largest and the most

distant object found in orbit around the sun since the discovery of Neptune

and its moon Triton in 1846. Eris is three times more distant, and takes more

than twice as long to orbit the sun, than the next closest plutoid, Pluto.

Accepted as a dwarf planet on September 13, 2006.

Makemake – discovered on March 31, 2005, is the third largest known dwarf

planet in the Solar System and about a third of the diameter of Pluto. Initially

known as 2005 FY9 (and later given the number 136472), Makemake is

named after a creator god of Rapa Nui (Easter Island). Accepted as a dwarf

planet on July 11, 2008.

Haumea: discovered on December 28, 2004 and named after the Hawaiian

goddess of fertility and childbirth, is a dwarf planet in the Kuiper belt one-third

the mass of Pluto. Although its shape has not been directly observed,

calculations suggest it is an ellipsoid, with its greatest axis twice as long as its

shortest. The two known moons are believed to have broken off the dwarf

planet during an ancient collision, and are thus named after two of Haumea's

daughters, Hiʻiaka and Nāmaka. Accepted as a dwarf planet on September

17, 2008.

No space probe has visited any of the dwarf planets. This will change if

NASA's Dawn and New Horizons robotic missions reach Ceres and Pluto,

respectively, in 2015 as planned. Dawn is also slated to orbit and observe

another potential dwarf planet, Vesta, in 2011.

The next three largest objects in the main asteroid belt – Vesta, Pallas,

and Hygiea – could eventually be classified as dwarf planets if it is shown that

their shape is determined by hydrostatic equilibrium. While uncertain, the present

data suggests that it is unlikely for Pallas and Hygiea. Vesta appears to deviate

from hydrostatic equilibrium only because of a large impact that occurred after it

solidified; the definition of dwarf planet does not specifically address this issue.

Many TNOs are thought to have icy cores and therefore would require a

diameter of 400 km – only about 3% of that of Earth – to relax into gravitational

equilibrium, making them dwarf planets of the plutoid class. Although only rough

estimates of the diameters of these objects are available, it is believed that up to

200 dwarf planets may be found when the entire region known as the Kuiper belt

is explored, and that the number might be as high as 2,000 when objects

scattered outside the Kuiper belt are considered.

The status of Charon (currently regarded as a satellite of Pluto) remains

uncertain, as there is currently no clear definition of what distinguishes a satellite

system from a binary (double planet) system. Charon could be considered a

planet because:

Charon independently would satisfy the size and shape criteria for a dwarf

planet status;

Charon revolves with Pluto around a common barycentre located in free

space between the two bodies (rather than inside of the system primary).

Apart Charon, a total of 18 known moons are massive enough to have

relaxed into a rounded shape under their own gravity. These bodies have no

significant physical differences from the dwarf planets, but are not considered

members of that class because they do not directly orbit the Sun. They are

Earth's moon, the four Galilean moons of Jupiter (Io, Europa, Ganymede, and

Calisto), seven moons of Saturn (Mimas, Enceladus, Tethys, Dione, Rhea, Titan,

and Iapetus), five moons of Uranus (Miranda, Ariel, Umbriel, Titania, and

Oberon), one moon of Neptune (Triton). All these bodies orbit around a centre of

gravity that is deep inside of their massive planet. However, there has been no

official recognition that the location of the barycenter is involved with the

definition of a satellite.