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 precise 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.
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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 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. 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 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 km. 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, km and an apoapsis altitude of 33, 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. 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.
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.
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Determining Orbital Elements. 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. 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. 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 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 ,. 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, t o , known as the "epoch": o , true anomaly at epoch , u o , argument of latitude at epoch , or o , true longitude at epoch. If the orbit is both circular and equatorial, o is simply the true angle from x to r o , both of which are always defined. 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.
Hyperbolic Departure and Approach. 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.http://yuzu-washoku.com/components/2020-08-17/1302.php
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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. Combining equations 4. 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.
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Arrival at the Target Planet As before, the relative velocity vector and magnitude are calculated using equations 5. If a dead center hit on the target planet is planned, then we solve the Gauss problem setting r 2 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.
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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.
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,. The angle is calculated as follows:. From the following we obtain the impact parameter, b. 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 r x , r y and r z are the scalar components of the planet's position vector, the components of the target point, r x ', r y ' and r z ', 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. 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. 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.
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