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.
A heliocentric orbit (also called circumsolar orbit) is an orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun.All planets, comets, and asteroids in the Solar System, and the Sun itself are in such orbits, as are many artificial probes and pieces of debris.The moons of planets in the Solar System, by contrast, are not in. Solar System Video showing the 8 planets of the Solar System orbiting the Sun. As we move out from Mercury, Venus, Earth and Mars, towards the gas giant plan.
Galileo discovered evidence to support Copernicus’ heliocentric theory when he observed four moons in orbit around Jupiter. Beginning on January 7, 1610, he mapped nightly the position of the 4 “Medicean stars” (later renamed the Galilean moons). Over time Galileo deduced that the “stars” were in fact moons in orbit around Jupiter. Topics similar to or like. Heliocentric orbit. Orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun. Fifth planet from the Sun and the largest in the Solar System. We would like to show you a description here but the site won’t allow us.
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.
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 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.
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 InjectionDV (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 InsertionDV (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:
Heliocentric Orbits
- 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.
Heliocentric Orbital Motion
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.
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).
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:
Compiled, edited and written in part by Robert A. Braeunig, 2012, 2013.
Bibliography