INTERPLANETARY FLIGHT |
- Heliocentric Transfer Orbit
- The Gauss Problem
- Determining Orbital Elements
- Hyperbolic Departure and Approach
- Gravitational Assist
An interplanetary spacecraft spends most of its flight time moving
under the gravitational influence of a single body – the Sun. Only for
brief periods, compared with the total mission duration, is its path
shaped by the gravitational field of the departure or arrival planet.
The perturbations caused by the other planets while the spacecraft is
pursuing its heliocentric course are negligible.
The computation of a precision orbit is a trial-and-error
procedure involving numerical integration of the complete equations of
motion where all perturbation effects are considered. For preliminary
mission analysis and feasibility studies it is sufficient to have an
approximate analytical method for determining the total V
required to accomplish an interplanetary mission. The best method
available for such analysis is called the patched-conic approximation.
The patched-conic method permits us to ignore the gravitational
influence of the Sun until the spacecraft is a great distance from the
Earth (perhaps a million kilometers). At this point its velocity
relative to Earth is very nearly the hyperbolic excess velocity.
If we now switch to a heliocentric frame of reference, we can determine
both the velocity of the spacecraft relative to the Sun and the
subsequent heliocentric orbit. The same procedure is followed in reverse
upon arrival at the target planet's sphere of influence.
The first step in designing a successful interplanetary
trajectory is to select the heliocentric transfer orbit that takes the
spacecraft from the sphere of influence of the departure planet to the
sphere of influence of the arrival planet.
If you have not already done so, before continuing it is recommended that you first study the Orbital Mechanics
section of this web site. It is also recommended, if you are not
already familiar with the subject, that you review our section on Vector Mathematics.
Heliocentric-Ecliptic Coordinate System
Our first requirement for describing an orbit is a suitable
inertial reference frame. In the case of orbits around the Sun, such as
planets, asteroids, comets and some deep-space probes describe, the
heliocentric-ecliptic coordinate system is convenient. As the name
implies, the heliocentric-ecliptic system has its origin at the center
of the Sun. The X-Y or fundamental plane coincides with the ecliptic,
which is the plane of Earth's revolution around the Sun. The
line-of-intersection of the ecliptic plane and Earth's equatorial plane
defines the direction of the X-axis. On the first day of spring a line
joining the center of Earth and the center of the Sun points in the
direction of the positive X-axis. This is called the vernal equinox
direction. The Y-axis forms a right-handed set of coordinate axes with
the X-axis. The Z-axis is perpendicular to the fundamental plane and is
positive in the north direction.
It is known that Earth wobbles slightly and its axis of rotation
shifts in direction slowly over the centuries. This effect is known as
precession and causes the line-of-intersection of Earth's equator and
the ecliptic to shift slowly. As a result the heliocentric-ecliptic
system is not really an inertial reference frame. Where extreme
precision is required, it is necessary to specify that the XYZ
coordinates of an object are based on the vernal equinox direction of a
particular year or epoch.
For transfers to most of the planets, we may initially consider that
the planetary orbits are both circular and coplanar. In the Orbital Mechanics
section of this site we discussed the problem of transferring between
coplanar orbits and found that the most economical method, from the
standpoint of V required, was the Hohmann transfer. A Hohmann transfer between Earth and Mars is pictured in Figure 5.1 below.
While it is generally desirable that the transfer orbit be
tangential to Earth's orbit at departure, it may be preferable to
intercept Mars' orbit prior to apogee, especially if the spacecraft is
to return to Earth. The Hohmann transfer, if continued past the
destination planet, would not provide a suitable return trajectory. For a
one-way trip this is irrelevant; however, for a probe that is to be
recovered or for a manned mission, this consideration is important. The
outbound trip to Mars on the Hohmann trajectory consumes between 8 and 9
months. If the spacecraft continued its flight it would return to the
original point of departure only to find Earth nearly on the opposite
side of its orbit. Therefore, either the spacecraft must loiter in the
vicinity of Mars for nearly 6 months or the original trajectory must be
modified so that the spacecraft will encounter Earth at the point where
it recrosses Earth's orbit.
If the spacecraft is to encounter the target planet at the time it
crosses the planet's orbit then obviously Earth and the target planet
must have the correct angular relationship at departure. The angle
between the radius vectors to the departure and arrival planets is
called ,
the phase angle at departure, and is pictured in Figure 5.2 for a Mars
trajectory. The total sweep angle from departure to arrival is just the
difference in the true anomaly at the two points, 2–1.
From Figure 5.2 we can see that the transfer is the one-tangent burn
type, which we examined previously. Selecting a transfer orbit allows
the determination of the change in true anomaly and the time-of-flight
using equations (4.67) and (4.71).
The target planet will move through an angle of t(t2–t1) while the spacecraft is in flight, where t is the angular velocity of the target planet. Thus, the correct phase angle at departure is,
The requirement that the phase angle at departure be correct severely
limits the times when a launch may take place. The heliocentric
longitudes of the planets are tabulated in The Astronomical Almanac, and these may be used to determine when the phase angle will be correct. Alternatively, the page Planet Positions
provides the data and demonstrates the methods necessary to estimate
planet positions without needing to refer to other sources.
Mars Transfer Trajectories
The methods described above provide only a very rough estimate of
the phase angle, particularly in the case of Mars. The orbit of Mars is
significantly eccentric, meaning its angular velocity changes
considerably depending on whether it is near perihelion or aphelion at
the time of transfer. For a better estimate we can no longer consider
the orbit to be circular.
As can be seen from Figure 5.2, the proper alignment for a
transfer to Mars occurs in the months just prior to an opposition. The
location of Mars within its orbit at the time of opposition depends on
the time of year the opposition occurs. Perihelion oppositions occur in
the August-September time period, and aphelion oppositions occur in the
February-March time period. We can, therefore, link the phase angle
required to the time of year that we initiate the transfer.
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Figure 5.3 above is a chart showing departure phase angle versus
departure calendar date. Four curves are shown, each representing a
different transfer orbit. The trajectories are tangent to Earth's orbit
at departure and differ in the number of degrees the spacecraft travels
around the Sun before intercepting Mars, i.e. the change in true
anomaly, .
If the interplanetary trajectory carries the spacecraft less than 180
degrees around the Sun, it's called a Type-I trajectory. If the
trajectory carries it 180 degrees or more around the Sun, it's called a
Type-II. A Type-I trajectory is pictured in Figure 5.2. In a Type-II
trajectory, the intercept point is at the second Mars orbit crossing.
Figure 5.3 gives the required phase angle for transfers
departing on the dates listed across the bottom of the chart. To use
Figure 5.3 it is necessary to find the date when the actual phase angle
equals the required angle obtained from the chart. For example, let's
say we are planning a mission to launch around the October-13 opposition
of the year 2020. It's decided we'll use a Type-I trajectory in which
the spacecraft's true anomaly change is 150 degrees (magenta curve). In
Table 1 we list the actual Mars-Earth phase angle for the months leading
up to the October opposition, along with the phase angle read from
Figure 5.3. We see that there is a date in July when the two numbers are
equal. We can interpolate that the departure date will be sometime in
the third week of July and the departure phase angle will be
approximately 30 degrees.
Just as phase angle is dependent on Mars' location within its
orbit, so is the time of flight. After estimating the departure date
from Figure 5.3, we can use Figure 5.4 below to estimate the flight
duration. For instance, for the July departure window determined above,
the time of flight for a trajectory with a true anomaly change of 150o
is found to be about 207 days. The letters superimposed on each curve
indicate the departure dates that will result in the spacecraft
intercepting Mars at perihelion (P), aphelion (A) or one of the two
nodes (N).
Referring again Figure 5.2, we see that the flight path angle of the
transfer orbit is positive at the first Mars orbit crossing and negative
at the second Mars orbit crossing. Therefore, it may be preferable to
use a Type-I trajectory when interception occurs with Mars in the part
of its orbit past perihelion and approaching aphelion, when the planet's
flight path angle is likewise positive. Conversely, a Type-II
trajectory may be preferable when interception occurs with Mars in the
part of its orbit past aphelion and approaching perihelion. Having Mars
and spacecraft flight path angles both positive or both negative reduces
the angle between the velocity vectors, and thus the relative velocity.
This can be critical when V is the limiting factor. Of course a Type-I trajectory is always preferable when minimizing flight time is most critical.
Non-coplanar Trajectories
Up to now we have assumed that the planetary orbits all lie in
the plane of the ecliptic. However, we know that all planets other than
Earth have orbits inclined to the ecliptic. A good procedure to use when
the target planet lies above or below the ecliptic at intercept is to
launch the spacecraft into a transfer orbit that lies in the ecliptic
plane and then make a simple plane change during mid-course when the
true anomaly change remaining to intercept is 90-degrees. This minimizes
the magnitude of the plane change required and is illustrated in Figure
5.4 below. Since the plane change is made 90o short of intercept, the required inclination is just equal to the ecliptic latitude, , of the target planet at the time of intercept, t2. The V required to produce a plane change was examined previously, and is calculated using equation (4.73).
Alternatively, the injection maneuver that places the spacecraft on
its interplanetary trajectory can include a plane change to correctly
orient the plane of the transfer orbit to intercept the target planet.
Such an orbit between Earth and Mars is pictured in Figure 5.6 below.
Since Earth lies in the ecliptic plane, the departure point defines one
of the transfer orbit's two nodes, with the other node 180 degrees away
on the opposite side of the Sun. Unless the target planet happens to
also be passing through one of its nodes at the time of interception, a
near 180-degree transfer is not possible without a prohibitively high
inclination. Intercepting the target with a true anomaly change several
degrees less than 180o (as pictured) or several degrees more than 180o can be achieved with a manageably low inclination.
On New Year's Day 1801, Giuseppe Piazzi of Italy observed for the
first time an object that turned out to be the minor planet Ceres. He
was only able to observe the asteroid for about one month before it was
lost in the glare of the Sun. The challenge of rediscovering Ceres when
it reappeared from behind the Sun seduced the intellect of young German
mathematician Carl Friedrich Gauss. Exactly one year later, on New
Year's Day 1802, Ceres was rediscovered precisely where the ingenious
and detailed calculations of Gauss had predicted she must be found.
The method that Gauss used is just as pertinent today as it was
in 1802, but for a different reason. The data that Gauss used to
determine the orbit of Ceres consisted of the right ascension and
declination at three observation times. His method is much simplified if
the original data consists to two position vectors and the
time-of-flight between them. The technique of determining an orbit from
two position vectors and time is of considerable interest to modern
astrodynamics since it has direct application in the solution of
intercept and rendezvous.
We may define the Gauss problem as follows: Given position vectors r1 and r2, the time of flight from r1 to r2, which we will call t, and the direction of motion, find velocity vectors v1 and v2. By "direction of motion" we mean whether the satellite is to go from r1 to r2 the "short way", through an angular change () of less than radians, or the "long way", through an angular change greater than radians. Obviously, there are an infinite number of orbits passing through r1 and r2, but only two that have the specified time-of-flight – one for each possible direction of motion.
One thing is immediately obvious from Figure 5.7; the two vectors r1 and r2 uniquely define the plane of the transfer orbit. If the vectors r1 and r2 are collinear and in opposite directions (=), the plane of the transfer orbit is not determined and a unique solution for v1 and v2 is not possible. If the two position vectors are collinear and in the same direction (=0 or 2), the orbit is a degenerate conic, but a unique solution is possible of v1 and v2.
The relationship between the four vectors r1, r2, v1 and v2 is contained in the f and g
expressions given below. It is not surprising, therefore, that nearly
every known method for solving the Gauss problem may be derived from the
f and g relations.
where,
As previously defined, is true anomaly, E is eccentric anomaly, a is semi-major axis, and GM is the standard gravitational parameter, while p is a geometrical constant of the conic section called the parameter or semi-latus rectum.
Consider equations (5.5), (5.6) and (5.7). There are seven variables — r1, r2, , t, p, a and E
— but the first four are known, so what we have is three equations in
three unknowns. The only trouble is that the equations are
transcendental in nature, so a trial-and-error solution is necessary. We
may outline the general method of solution as follows:
- Guess a trial value for one of the three unknowns, p, a or E directly or indirectly by guessing some other parameter of the transfer orbit that in turn establishes p, a or E.
- Use equations (5.5) and (5.7) to compute the remaining two unknowns.
- Test the result by solving equation (5.6) for t and check it against the given value of time-of-flight.
- If the computed value of t does not agree with the given value, adjust the trial value of the iteration variable and repeat the procedure until it does agree.
The last step is perhaps the most important of all, since the method
used to adjust the trial value of the iteration variable is what
determines how quickly the procedure converges to a solution. There are
several methods for solving the Gauss problem, however the only one we
will present here is the p-iteration technique. The method consists of
guessing a trial value of p from which we can compute a and E.
The first step in the solution is to find an expression for a as a function of p
and the given information. We will find it convenient to define three
constants that may be determined from the given information:
where r1 = |r1| and r2 = |r2|.
Once p is specified, a unique value of a is determined from
Once we have selected a trial value of p and computed a from equation (5.12), we are ready to solve for t and check it against the given time-of-flight. First, however, we need to determine E, or F in case a is negative (i.e. the orbit is hyperbolic).
From the trail value of p and the known information, we can compute f, g and from equations (5.5), (5.6) and (5.7). If a is positive, we can determine E from equations (5.5) and (5.7), rearranged as follows:
If a is negative, the corresponding f and g expressions involving F yield the following. Since we always assume F is positive, there is no ambiguity in determining F from this one equation.
The time-of-flight may now be determined from equation (5.16) or the corresponding equation involving F:
The limiting values of p correspond to two parabolic orbits passing through r1 and r2. The values of p that specify the parabolic orbits we'll call pi and pii. For less than radians, p must lie between pi and infinity. For greater than radians, p must lie between 0 and pii. Since it is important that the first trial value, as well as all subsequent guesses for p, lie within the prescribed limits, we should first compute pi or pii.
The method used to adjust the trial value of p to give the desired time-of-flight is crucial in determining how rapidly p
converges to a solution. Several simple methods may be used
successfully, such as the bisection or linear interpolation techniques.
In the bisection method we must find two trial values of p, one that gives too small a value for t
and one that gives too large a value. The solution is then bracketed
and, by choosing our next trial value half way between the first two, we
can keep it bracketed while reducing the interval of uncertainty to
some arbitrarily small value.
In the linear interpolation method we choose two trial values of p, which we will call pn-1 and pn. If tn-1 and tn are the times-of-flight corresponding to these trial values of p, then we select a new value from,
This scheme can be repeated, always retaining the latest two trial values of p
and their corresponding times-of-flight for use in computing a still
better trial value from equation (5.20). It is not necessary that the
initial two trial values bracket the answer.
We can summarize the steps involved in solving the Gauss problem via the p-iteration technique as follows:
- Evaluate the constants k, and m from r1, r2 and using equations (5.9) through (5.11).
- Determine the limits on the possible values of p by evaluating pi and pii from equations (5.18) and (5.19).
- Pick a trial value of p within the appropriate limits.
- Using the trial value of p, solve for a from equation (5.12). The type conic orbit will be known from the value of a.
- Solve for f, g and from equations (5.5), (5.6) and (5.7).
- Solve for E or F, as appropriate, using equations (5.13) and (5.14) or equation (5.15).
- Solve for t from equation (5.16) or (5.17) and compare it with the desired time-of-flight.
- Adjust the trial value of p using one of the iteration methods discussed above until the desired time-of-flight is obtained.
- Evaluate from equation (5.8) and then solve for v1 and v2 using equations (5.3) and (5.4).
The p-iteration method converges in all cases except when r1 and r2 are collinear. Its main disadvantage is that separate equations are used for the ellipse and hyperbola.
Click here for example problem #5.3
Click here for example problem #5.4
Selecting a Transfer Orbit
Each time the Gauss problem is solved, the result gives just one
of an infinite number of possible transfer orbits. It was previously
stated that it generally desirable that the transfer orbit be tangential
to Earth's orbit at departure. This is true only in that it minimizes
the V
required to inject the spacecraft into its transfer orbit; however, it
likely results in a less than optimum condition at target intercept. A
one-tangent burn produces a trajectory that crosses the orbit of the
target planet with a relatively large flight path angle, resulting in a
large relative velocity between the spacecraft and planet. This relative
velocity can be significantly reduced by selecting a transfer orbit
that reduces the angle between the velocity vectors of the spacecraft
and target at the moment of intercept. Improving the intercept condition
(1) increases the duration of a close flyby encounter, (2) reduces the V required for orbit insertion, or (3) lowers the spacecraft's velocity at atmospheric entry.
Tables 2 and 3 below provide sample data for a hypothetical mission to Mars in the year 2020. Table 2 gives the V
required for Trans-Mars Injection (TMI) for a variety of different
departure dates and times of flight. TMI is the maneuver that places the
spacecraft into a trajectory that will intercept Mars at the desired
place and time. In this sample, it is assumed that TMI is performed from
an Earth parking orbit with an altitude of 200 km. Table 3 gives the V
required for Mars-Orbit Insertion (MOI) for the same departure dates
and times of flight found in Table 2. MOI, as it names implies, is the
maneuver that slows the spacecraft to a velocity that places it into the
desired orbit around Mars. In this sample, it is assumed that MOI is
performed at periapsis of a insertion orbit with a periapsis altitude of
1,000 km and an apoapsis altitude of 33,000 km. Placing a spacecraft
into a high eccentricity orbit such as this is common, as it provides
for a MOI burn with a relatively low V.
Table 2 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Trans-Mars Injection DV (m/s), launch altitude = 200 km | |||||||||||
Departure Date, 2020 |
Time of Flight (days) | ||||||||||
180 | 185 | 190 | 195 | 200 | 205 | 210 | 215 | 220 | 225 | 230 | |
7/7 | 3876 | 3862 | 3854 | 3851 | 3853 | 3863 | 3881 | 3912 | 3962 | 4043 | 4180 |
7/12 | 3841 | 3830 | 3824 | 3823 | 3826 | 3835 | 3851 | 3877 | 3917 | 3978 | 4074 |
7/19 | 3819 | 3812 | 3808 | 3808 | 3811 | 3819 | 3833 | 3853 | 3882 | 3925 | 3988 |
7/26 | 3834 | 3829 | 3826 | 3826 | 3829 | 3836 | 3846 | 3862 | 3883 | 3913 | 3956 |
8/2 | 3892 | 3887 | 3885 | 3884 | 3886 | 3890 | 3897 | 3908 | 3923 | 3943 | 3972 |
8/9 | 3999 | 3994 | 3990 | 3987 | 3987 | 3987 | 3991 | 3996 | 4005 | 4017 | 4034 |
8/16 | 4162 | 4154 | 4147 | 4141 | 4137 | 4133 | 4131 | 4131 | 4133 | 4138 | 4146 |
8/23 | 4386 | 4373 | 4362 | 4351 | 4341 | 4332 | 4325 | 4318 | 4313 | 4310 | 4309 |
Table 3 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Mars Orbit Insertion DV (m/s), insertion orbit = 1000 × 33000 km | |||||||||||
Departure Date, 2020 |
Time of Flight (days) | ||||||||||
180 | 185 | 190 | 195 | 200 | 205 | 210 | 215 | 220 | 225 | 230 | |
7/7 | 1371 | 1258 | 1163 | 1086 | 1025 | 982 | 959 | 957 | 984 | 1052 | 1187 |
7/12 | 1290 | 1188 | 1102 | 1033 | 979 | 940 | 918 | 915 | 933 | 982 | 1074 |
7/19 | 1186 | 1097 | 1024 | 965 | 920 | 888 | 870 | 866 | 879 | 911 | 970 |
7/26 | 1093 | 1019 | 957 | 909 | 872 | 847 | 833 | 830 | 840 | 864 | 905 |
8/2 | 1016 | 954 | 904 | 865 | 837 | 818 | 808 | 808 | 817 | 836 | 867 |
8/9 | 957 | 907 | 868 | 838 | 817 | 804 | 799 | 801 | 811 | 828 | 853 |
8/16 | 920 | 881 | 852 | 830 | 816 | 809 | 808 | 813 | 823 | 839 | 862 |
8/23 | 910 | 881 | 860 | 846 | 838 | 836 | 839 | 846 | 857 | 873 | 893 |
As can be seen from Tables 2 and 3, in most instances, TMI V and MOI V
are inversely proportional. That is, trying to optimize one increases
the other, and vice versa. Selecting the "best" transfer orbit therefore
comes down to making a compromise. The size of the launch window is
also often limited by the V
budget. For example, suppose our launch vehicle can deliver no more
than 3,900 m/s for TMI, and our spacecraft's MOI budget is 900 m/s. Our
potential launch opportunities are limited to those in which both of
these conditions are met, which we see represented by the launch dates
and flight durations highlighted above.
Solving the Gauss problem gives us the position and velocity vectors, r and v, of a spacecraft in a heliocentric-ecliptic orbit. From these vectors we can determine the six orbital elements that describe the motion of the satellite. The first step is to form the three vectors, h, n and e, illustrated in Figure 5.08.
The specific angular momentum, h, of a satellite is obtained from
It is important to note that h is a vector perpendicular to the plane of the orbit.
The node vector, n, is defined as
From the definition of a vector cross product, n must be perpendicular to both z and h. To be perpendicular to z, n would have to lie in the ecliptic plane. To be perpendicular to h, n would have to lie in the orbital plane. Therefore, n must lie in both the ecliptic and orbital planes, or in their intersection, which is called the "line of nodes." Specifically, n is a vector pointing along the line of nodes in the direction of the ascending node. The magnitude of n is of no consequence to us; we are only interested in its direction.
The third vector, e, is obtained from
Vector e points from the center of the Sun (focus of the
orbit) toward perihelion with a magnitude exactly equal to the
eccentricity of the orbit.
Now that we have h, n and e we can preceed rather easily to obtain the orbital elements. The semi-major axis, a, and the eccentricity, e, follow directly from r, v, and e,
while all the remaining orbital elements are simply the angles between
two vectors whose components are now known. If we know how to find the
angle between two vectors the problem is solved. In general, the cosine
of the angle, , between two vectors a and b is found by dividing the dot product of the two vectors by the product of their magnitudes.
Of course, being able to evaluate the cosine of an angle does not
mean that we know the angle. We still have to decide whether the angle
is smaller or greater than 180 degrees. The answer to this quadrant
resolution problem must come from other information in the problem as we
shall see.
We can outline the method of finding the orbital elements as follows:
- Calculate a and e,
- Since i is the angle between z and h,
- (Inclination is always less than 180o)
- Since is the angle between x and n,
- (If ny > 0 then is less than 180o)
- Since is the angle between n and e,
- (If ez > 0 then is less than 180o)
- Since o is the angle between e and r,
- (If r • v > 0 then o is less than 180o)
- Since uo is the angle between n and r,
- (If rz > 0 then uo is less than 180o)
- Calculate and o,
- ( and o are always less than 360o)
The angle , longitude of periapsis,
is sometimes used in place of argument of periapsis. As a substitute
for the time of periapsis passage, any of the following may be used to
locate the spacecraft at a particular time, to, known as the "epoch": o, true anomaly at epoch, uo, argument of latitude at epoch, or o, true longitude at epoch.
If there is no periapsis (circular orbit), then is undefined, and o = + uo. If there is no ascending node (equatorial orbit), then both and uo are undefined, and o = + o. If the orbit is both circular and equatorial, o is simply the true angle from x to ro, both of which are always defined.
The procedure outlined above describes a spacecraft in a solar orbit,
but the method works equally well for satellites in Earth orbit, or
around another planet or moon, where the position and velocity vectors
are known in the geocentric-equatorial reference plane. Note, however,
that it is customary for the geocentric-equatorial coordinate system to
use unit vectors i, j and k instead of x, y and z as used in the heliocentric-ecliptic system.
Solving the Gauss problem gives us the heliocentric transfer orbit
that takes the spacecraft from the sphere of influence of the departure
planet to the sphere of influence of the arrival planet. To complete the
patched-conic method, we will now examine the trajectories within the
planets' spheres of influence. For this discussion we will assume the
departure planet is Earth, though the methods described apply equally to
other departure planets.
Escape from Earth's Sphere of Influence
Once the heliocentric transfer orbit has been selected, we next
determine the spacecraft's velocity relative to Earth. The relative
velocity, which we will define as the vector vs/p, is the difference between the spacecraft's heliocentric velocity, vs, and the planet's orbital velocity, vp (see Figure 5.10).
where the vector magnitude is,
We can now proceed to establish the injection or launch conditions
near the surface of Earth that will result in the required hyperbolic
excess velocity. Since Earth's sphere of influence has a radius about
one million kilometers, we assume that V∞ ≈ Vs/p. Combining equations (4.78) and (4.88), we can solve for the injection velocity, Vo, as follows:
If launch occurs from an Earth parking orbit, the injection V is simply the injection velocity minus the orbital velocity,
The hyperbolic excess velocity is extremely sensitive to small errors
in the injection velocity. For a Hohmann transfer to Mars, a one
percent error in injection velocity results in a 15 percent error in
hyperbolic excess velocity.
The departure asymptote should be parallel to the hyperbolic
excess velocity, as shown in Figure 5.11. The zenith angle of the
asymptote, , is obtained from the dot product r • vs/p.
To determine the further parameters of the hyperbolic escape trajectory, please refer to the hyperbolic orbit as previously examined.
Arrival at the Target Planet
As before, the relative velocity vector and magnitude are calculated using equations (5.33) and (5.34).
If a dead center hit on the target planet is planned, then we solve the Gauss problem setting r2
equal to the position vector of the planet at arrival. This ensures
that the target planet will be at the intercept point at the same time
the spacecraft is there. It also means that the relative velocity
vector, upon arrival at the target planet's sphere of influence, will be
directed toward the center of the planet, resulting in a straight line
hyperbolic approach trajectory.
If it is desired to fly by the planet instead of impacting it,
then the transfer trajectory must be modified so that the spacecraft
crosses the target planet's orbit ahead of or behind the planet. If the
spacecraft crosses the planet's orbit a distance d from the planet, then the velocity vector vs/p, which represents the hyperbolic excess velocity on the approach hyperbola, is offset a distance b from the center of the target planet, as shown in Figure 5.12.
The sign of d is chosen depending on whether the
spacecraft is to cross ahead of (positive) or behind (negative) the
target planet. Assuming the target point lies within the same X-Y plane
as the planet, the rectangular components of d are,
where rx and ry are scalar components of the planet's position vector.
The angle is calculated as follows:
From the following we obtain the impact parameter, b
Recalling that V∞ ≈ Vs/p, we calculate the hyperbola's semi-major axis and eccentricity as follows:
To calculate the remaining parameters of the hyperbolic approach trajectory, see the hyperbolic orbit.
For a close flyby, and understanding that the patched-conic
method is only an approximate solution, it is generally adequate to
ignore the miss distance, d, when solving the Gauss problem,
assuming the position vector at arrival is equal to that of the planet.
However, if the flyby distance is large, an improved Gauss solution is
obtained by modifying the position vector to account for miss distance.
If rx, ry and rz are the scalar components of the planet's position vector, the components of the target point, rx', ry' and rz', are as follows:
In planning certain types of trajectories of spacecraft within the
solar system, engineers rely on a technique called gravitational assist
(also gravity assist, slingshot, or swing-by). This technique underlies
the feasibility of effecting a net change in both the speed and
direction of motion of a spacecraft by passage through the gravitational
field of a planet or a planetary satellite, typically in order to save
propellant, time, and expense.
A gravitational assist around a planet changes a spacecraft's
velocity relative to the Sun by entering and leaving the gravitational
field of a planet. The spacecraft accelerates as it approaches the
planet and decelerates while escaping its gravitational pull. Because
the planet orbits the Sun, this motion affects the spacecraft during the
maneuver. To accelerate, the spacecraft flies across the trailing side
of the planet, taking a small amount of the planet's orbital energy (as
pictured in Figure 5.13). To decelerate, the spacecraft flies across the
leading side of the planet. The sum of the kinetic energies of both
bodies remains constant. A gravitational assist can therefore be used to
change the spaceship's trajectory and speed relative to the Sun.
The resulting increase, or decrease, in the kinetic energy of the
spacecraft appears to contradict the casual expectation that in such an
encounter the kinetic energy of the spacecraft after the encounter
would be the same as that before the encounter. However, the energy
gained by the spaceship is equal in magnitude to that lost by the
planet, though the planet's enormous mass compared to the spacecraft
makes the resulting change in its speed negligibly small. These effects
on the planet are so slight that they can be ignored in the calculation.
Figure 5.13 shows the motion of a spacecraft relative to a planet during
a gravity assist maneuver. Encounters in space require the
consideration of three dimensions; however, an approximate solution to
the gravitational assist problem can be found using a simplified
two-dimensional model. The following conditions are assumed:
- Orbits of planet and spacecraft are coplanar.
- Y-axis is parallel to the planet's position vector, positive outward from Sun.
- X-axis is in the orbital plane normal to the Y-axis, positive in the prograde direction.
- Planet's velocity (Vp) and flight path angle (p) are given.
- Spacecraft's initial velocity (Vsi), flight path angle (si), and miss distance (d) are given.
The planet's velocity vector and the spacecraft's initial velocity vector are solved for using,
The initial relative velocity vector and its magnitude are solved for using,
(Recall that V∞ ≈ Vs/p)
The angle i is obtained as follows (making sure to check quadrant),
The impact parameter, b, semi-major axis, a, and eccentricity, e, are solved for using equations (5.40), (5.41) and (5.42).
The turning angle, , is solved for using equation (4.80), and the angle f is calculated as follows,
The final relative velocity vector is,
And the spacecraft's final velocity vector and magnitude are,
Finally, we calculate the spacecraft's final flight path angle as follows,
Compiled, edited and written in part by Robert A. Braeunig, 2012, 2013.
http://www.braeunig.us/space/index.htm
Penulis : Drs.Simon Arnold Julian Jacob
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