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Transcript of · Web viewAt aphelion the spacecraft, which has the velocity in the "fixed" solar frame, is...
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.