Pluto New Horizons

Patrick Hunkins

December 15, 2005

1. Introduction

2. Literature Search

3. Trajectory Analysis and Commentary

4. Extension

5. Recommendation

6. Conclusions

Abstract:

The aim of Pluto New Horizon’s mission is to perform the first ever reconnaissance of Pluto and its moon Charon, and then journey into the previously unexplored Kuiper Belt to perform one or two close fly-bys of Kuiper Belt Objects. While at Pluto, the New Horizons Orbiter will aim to map the surface composition and geology of Pluto and Charon, characterize the atmosphere of Pluto and its escape rate, look for an atmosphere on Charon, and search for rings and other satellites in the Pluto Charon System. The mission is scheduled to launch in January of 2006 with arrival at Pluto in July 2015. The satellite will perform a gravity assist at Jupiter before coasting toward the Pluto Charon System. This paper will provide a thorough analysis of the astrodynamics involved in getting the satellite to Pluto and beyond within the planned time frame. Additionally, this paper will propose an extension to the mission that involves additional close planetary fly-bys of Uranus, Neptune, or both, similar to the planned Jupiter fly-by, before heading out into the Kuiper Belt. This extension would eliminate Pluto as a destination. It is a common belief that Pluto is a large Kuiper Belt Object rather than a planet, so this extension would essentially add planetary science while eliminating the reconnaissance of one Kuiper Belt Object, but still allowing Kuiper Belt Object fly-bys. A thorough analysis of this extension will be presented along with recommendations based on this analysis.

1. Introduction:

The goal of the Pluto New Horizons mission is to perform the first ever reconnaissance of Pluto and its moon Charon, before venturing out into the Kuiper Belt, a large region of small icy objects that orbit the Sun beyond the orbit of Neptune. Our solar system consists of three distinct regions: the rocky inner planets, the gas giants, and the icy dwarfs at the outskirts of the solar system. To this point, there have been no missions that have visited this third region of the solar system. For this reason, our knowledge of the planets is incomplete. This mission will aim to fill those gaps in our knowledge by providing data about this third class of objects orbiting the Sun. The goals of the mission are to: 1) Map the surface composition of Pluto and Charon, 2) Characterize geology and morphology of Pluto and Charon, 3) Characterize the neutral atmosphere of Pluto and its escape rate, 4) Search for an atmosphere around Charon, 5) Map the surface temperatures of Pluto and Charon, 6) Search for rings and additional satellites around Pluto, and 7) Conduct similar investigations into one or more Kuiper Belt Objects.

The spacecraft itself will carry seven science instruments to accomplish the mission goals. The satellite with instruments labeled is shown in Figure 1. These instruments are described in Table 1.

Figure 1

Science Instrument	Function
Ralph (Multispectral Visible Imaging Camera)	Take high-resolution color and surface composition maps of Pluto and Charon.
Alice (Ultra-Violet Mapping Spectrometer)	Gain information on the composition of Pluto’s atmosphere.
REX (Radio Science Experiment)	Communication between the spacecraft and Earth.
LORRI (Long Range Reconnaissance Imager)	Used to take images with a 100-meter resolution at the closest pass between Pluto and the spacecraft.
SWAP (Solar Wind Around Pluto: Low Energy Plasma Instrument)	Used to measure solar wind. Will determine whether or not Pluto has a magnetosphere and at what rate the atmosphere on Pluto is escaping.
PEPSSI (Pluto Energetic Particle Spectrometer Science Investigation: Medium Energy Particle Spectrometer)	Search for atoms that escape Pluto’s atmosphere and are subsequently charged by the solar wind.
SDC (Student Dust Counter)	Count and measure the size of dust particles encountered during the entire trip.

Table 1

The New Horizons is scheduled to launch from Cape Canaveral on January 11, 2006 with a two hour long launch window that opens at 19:07 UT. If the New Horizons is unable to launch on this day, there will be other opportunities for it to launch continuing until February 14. At this point, the window will not open again until 2007. The satellite will launch on a Lockheed Martin built Atlas V 551 with a Star 48B third stage. After leaving Earth, the spacecraft will coast on its way to Jupiter where it will perform a gravity assist maneuver. This gravity assist will occur from February 25 to March 2, 2007. During this time, the spacecraft will come within roughly 32 Jovian radii from Jupiter. After the gravity assist, the spacecraft will coast on its way to Pluto with arrival at Pluto occurring on July 14, 2015. During its pass of Pluto, the New Horizons spacecraft will come as close as 9600 km to Pluto and 27000 km to Charon at a relative velocity between Pluto and the spacecraft of 11 km/s. Figure 2 illustrates the planned trajectory.

Figure 2

The velocity of the satellite when leaving Earth for the extension should be similar to the velocity leaving Earth for the planned mission in order to ensure that the extension is feasible from a launch capability standpoint. The Jupiter gravity assist will be conducted in a similar manner, but instead of traveling to Pluto, the spacecraft will set a trajectory to Uranus or Neptune in order to conduct some planetary science observations before heading off into the previously unexplored Kuiper Belt.

This paper will examine the astrodynamics involved in flying along the proposed trajectory. In order to do this, each aspect of the trajectory will be examined separately, with the ending conditions of the previous step representing the initial conditions of the subsequent step. All equations used to analyze the problem can be derived from the equation that governs two-body motion.

In this equation, represents the vector from the central body that is being orbited to the satellite, and represents the gravitational parameter of the body that is being orbited. All calculations pertaining to orbit maneuvers will assume that the maneuvers are carried out with an impulsive .

In this analysis, the assumption will be that the only force acting on the satellite during its trek to Pluto is the force of gravity. Additionally, the analysis will assume that the satellite is never subjected to gravity forces from more than one body. For instance, when the satellite gets close enough to Jupiter, the force of gravity from the Sun will be ignored, and only the gravity of Jupiter will be used to propagate the trajectory. When the satellite again moves far enough away from Jupiter, Jupiter’s gravity will be ignored, and the Sun’s gravity will again be considered. Additionally, it will be assumed that each body can be approximated as a point mass, and any perturbations caused by deviations from this ideal gravity field will be ignored.

2. Literature Search:

Websites

pluto.jhuapl.edu

nssdc.gsfc.nasa.gov/database/MasterCatalog?sc=NHORIZONS

www.space.com/scienceastronomy/pluto_wait_030227.html

http://www.answers.com/topic/new-horizons

http://www.ifa.hawaii.edu/faculty/jewitt/kb.html

http://aa.usno.navy.mil/data/docs/EarthSeasons.html

http://en.wikipedia.org/wiki/Pluto

http://www.heavens-above.com

Books

Vallado, David: Fundamentals of Astrodynamics and Applications, Second Edition. Microcosm Press, El Segundo, CA; 2001.

3. Trajectory Analysis and Commentary:

Achieving a thorough understanding of a planned trajectory is a vital portion of any satellite mission. The trajectory must be understood in order to design hardware (electrical energy sources, propulsion systems, etc.) that will satisfy the mission requirements. Understanding what will happen to the space craft under the influence of the various celestial bodies that it passes by will allow mission designers to optimize many aspects of the mission in order to save fuel, money, and time when necessary. This understanding is especially important in the case of the Pluto New Horizons Mission as the mission has a very long duration and will traverse half of the solar system. Due to this long travel time, and the temporal constraints imposed by the fact that Pluto is losing its atmosphere, many aspects of the satellite need to be optimized. Even with one of the most powerful rockets available (the Lockheed Martin Atlas V 551) the satellite needs to be very light if it hopes to achieve a large enough velocity to carry out the mission as planned.

A thorough understanding of the problems that will be faced from an astrodynamics perspective for interplanetary missions is vital in the aerospace climate today. There is a great deal of emphasis placed by NASA on the exploration of the solar system and seeking out its origins. Many of these missions involve visitation of celestial bodies throughout the solar system by spacecraft. Without a thorough knowledge of the dynamics involved in interplanetary travel, it is impossible to carry out this vision.

The analysis of the trajectory for the mission can be accomplished using the method of patched conics. Using this method, one will assume two-body motion between the satellite and the body in whose sphere of influence the satellite lies. All perturbations due to gravity from other bodies will be considered negligible. The problem of solving for the orbit that will get the New Horizons spacecraft to Pluto in the desired time frame is an example of Lambert’s Problem. One is given the desired times at which the spacecraft will launch, complete its Jupiter gravity assist, and end up arriving at the final destination. Given two positions and the amount of time between them, one can use a solution to Lambert’s Problem to determine the orbit between these two points. The stipulation is that for a valid trajectory, the velocity at the end of the Earth to Jupiter portion of the mission and the initial velocity of the Jupiter to Pluto portion of the mission must be related such that there is a valid hyperbolic orbit about Jupiter in between the two.

In order to perform a correct analysis, one must know the position of each of the planets in a Sun centered inertial coordinate system. This can be determined from the orbital elements. The orbital elements for the Earth, Jupiter, and Pluto are shown in Table 2.

	Earth	Jupiter	Pluto
a	149598023 km	778298361 km	5915799000 km
e	0.016708617	0.048494851	0.249050
i	0°	1.30326966°	17.14216667°
	N/A	100.46444064°	110.29713889°
	N/A	273.8668686°	113.8377222°

Table 2

Since the inclination of Earth’s orbit is 0°, the right ascension of the ascending node and argument of periapse are undefined. For this case, the true longitude is given as 100.46644851°. The true longitude is defined as the angle eastward from the vernal equinox to the eccentricity vector which points toward periapse. Now given the true anomaly at a given time, one can calculate the position and velocity at that time. In order to determine the true anomaly at any given time, one must know the time since periapse. For the launch of the satellite, it is of interest to know the time since periapse for Earth at the launch time. For the gravity assist, it is of interest to know the time since periapse for Jupiter at the time of closest pass of the Jupiter fly-by. Finally, it is necessary to know the time since periapse for Pluto at the time of closest approach to Pluto. The time of last periapse, the desired rendezvous time, and the time since periapse at the time of interest are shown in Table 3.

	Last Periapse	Time of Interest	Time Since Periapse
Earth	Jan 4 15:00 UTC, 2006	Jan 11 19:07 UTC, 2006	619620 sec.
Jupiter (Gravity Assist)	May 20, 1999	Feb 28, 2007	7.945 years
Pluto	September 5, 1989	July 14, 2015	25.855 years

Table 3

In order to extract true anomaly from the time since periapse, one needs to calculate the mean anomaly. This can be calculated from the mean motion. The mean motion can be calculated using the following equation:

(1)

In this equation, n represents the mean motion. The gravitational parameter for the Sun equals Substituting the semi-major axis into (1) one obtains:

The mean anomaly at the times of interest for the planets can then be calculated using the following equation:

(2)

This gives the following mean anomalies for the planets at the times of interest shown in the tables:

The mean anomaly can then be converted to the eccentric anomaly using the following relationship:

(3)

This equation obviously cannot be solved explicitly for the eccentric anomaly, so the Newton Raphson iteration technique must be used. This technique yields the following results for the eccentric anomalies:

In order to find the position of these planets at the times of interest, the eccentric anomalies must be converted to true anomalies. The eccentric anomaly can be converted to the true anomaly using the following equation:

(4)

In equation (4), represents the true anomaly. Solving this equation yields the following true anomalies for the planets at the times of interest.

Now that all of the orbital elements are known for the planets at the times of interest, it is possible to compute the position and velocity of these planets in the Sun centered inertial coordinate system. In order to do this, one must first compute the position and velocity of the planets in the perifocal system. This is a system in which the “P” axis points towards the periapse of the orbit, the “Q” axis points normal to the orbit plane, and the “W” axis obeys the right hand rule with respect to the other axes. In order to compute this velocity, the semi-parameter for each of the orbits must be calculated. The semi-parameter can be defined as:

(5)

The position and velocity in the perifocal system can then be calculated from the following equations:

These vectors can be converted to vectors in the inertial frame by performing the following matrix multiplications:

(8)

In the above equation, ROT3 represents a rotation about the Z-axis, and ROT1 represents a rotation about the X-axis. The previous relation obviously does not hold when considering the orbit of the Earth due to the fact that this orbit has zero inclination. The position and velocity of the Earth can be computed by simply taking the vectors in the perifocal system and rotating them about the Z-axis by the true longitude multiplied by negative one. The position and velocity of each of the planets at the time of interest is shown in Table 4.

	(km)	(km)	(km)	(km/s)	(km/s)	(km/s)
Earth	-44917931.81	140093238.61	0	-28.856	-9.186	0
Jupiter	-266450159.88	-752170218.34	9068977.14	12.163	-3.753	-0.257
Pluto	1206389908.91	-4777726630.73	162199170.02	5.373	0.217	-1.578

Table 4

The analysis will begin by looking at the interplanetary coast phase between Jupiter and Pluto. This problem is a classic application of Lambert’s Problem. In this problem, we have two positions and the time of flight between the two positions, and wish to determine the orbit that passes between the two points in the desired amount of time. This analysis will use the Universal Variable Solution to Lambert’s Problem to solve for the velocity required when leaving Jupiter to arrive at Pluto within the given time frame. The algorithm for computing this velocity is as follows.

Begin by computing the difference in true anomaly between the two positions. This can be done using the following equation:

(9)

Then compute a quantity “A” using the following equation:

(10)

Then pick an arbitrary initial value for the variable . In this case was chosen to be zero initially. Upper and lower bounds must also be chosen for . In this case, an upper bound of and a lower bound of were chosen. The variables and can be calculated from the value of using the following equations:

(11), (12)

Or for negative values of , the alternate equations are as follows:

(13), (14)

One can then use these variables to compute the following quantities:

If the computed value of is less than the desired time of flight, the lower bound on should be changed to the current value of . Otherwise the upper bound on should be changed to the current value of . A new value of should then be computed as the average of the upper and lower bounds. A new value of and will then need to be computed as well. New values of , , and should then be computed. This process should be repeated until the value of is reasonably close to the desired time of flight. For this problem, a maximum allowable difference of 0.000001 seconds was used. Once this problem has converged, the following equations can be used to obtain the initial and final velocity of the orbit between the two points:

The desired time of flight between Jupiter and Pluto in this problem is 8.372 years or 264192994 seconds. When the positions of Jupiter and Pluto shown in Table 4 along with the desired time of flight are plugged into the algorithm for the Universal Variables Solution to Lambert’s Problem, the following is obtained as the velocity in the Sun centered inertial system required when leaving Jupiter to arrive at Pluto at the desired time.

This will result in a velocity upon arrival at Pluto in the Sun centered inertial system of:

These velocities correspond to a hyperbolic escape trajectory with respect to the Sun.

The focus of the analysis will now shift to the transfer from Earth to Jupiter. Again, we are faced with a problem in which the desired position is known at two separate times, and we are trying to find the orbit that fits this. Again, this is an example of Lambert’s Problem and the Universal Variables Solution will be used to determine the velocity required when leaving Earth to arrive at Jupiter in the prescribed amount of time. When the Universal Variables Solution to Lambert’s Problem is used with a time of travel between the two positions of 35703936 seconds, one obtains the following velocity vector for the spacecraft when leaving Earth in the Sun centered inertial reference frame.

This will result in a velocity upon arrival at Jupiter in the Sun centered inertial system of:

This is again a hyperbolic escape trajectory with respect to the Sun.

The focus of the analysis will now shift to the Jupiter gravity assist itself. In order to use Jupiter to boost its velocity on the way to Pluto, the New Horizons must follow a hyperbolic trajectory about Jupiter. A picture of this type of trajectory is shown in Figure 3.

Figure 3

In order to analyze this portion of the mission, the velocity of the New Horizons with respect to Jupiter must be known. The following equation calculates the velocity of the New Horizons with respect to Jupiter at the point where it leaves Jupiter as well as when it arrives at Jupiter.

Defining as the velocity of the satellite when it reaches the edge of the sphere of influence of the planet, it can be seen then that it is necessary that for the New Horizons gravity assist about Jupiter in order to reach Pluto as planned. The spacecraft is to come within 32 Jovian radii of the planet during this fly-by, which means that for this maneuver should equal . The semi-major axis for a hyperbolic orbit can be computed using the following equation:

(23)

For Jupiter, , so . The eccentricity of the orbit can then be calculated using equation (24):

(24)

The eccentricity calculated for this orbit is 7.830. The angle as seen in Figure 3 can be calculated from the following relationship:

(25)

This gives a value of equal to 14.6749°. This means that the velocity of the New Horizons on its way to Jupiter will be at an angle of 165.3251° to the velocity as it leaves Jupiter. It can be seen that the velocity of the New Horizons at the point at which it arrives at Jupiter is slightly off from what it needs to be in both magnitude and orientation for the desired hyperbolic trajectory. Because of this, the New Horizons must perform a maneuver upon entering the sphere of influence of the planet in order to follow the desired path. The spacecraft must perform a such that the magnitude of its velocity is , it has an angle with the exiting velocity vector of 14.6749°, and lies in the same plane as the exiting velocity vector and the velocity vector at arrival before the maneuver. This can be explained mathematically with the following equations:

In these equations, is the velocity of the satellite relative to Jupiter prior to the maneuver, is the velocity of the satellite after completing the maneuver, and is the velocity of the satellite with respect to Jupiter as it leaves Jupiter’s sphere of influence. It can be shown that the velocity vector that satisfies these equations is:

This results in:

This impulsive maneuver will allow the New Horizons to follow the correct hyperbolic trajectory about Jupiter to send it on its way to Pluto.

In summary, the New Horizons spacecraft will follow a hyperbolic trajectory with respect to the Sun on its way from Earth to Jupiter, leaving on January 11, 2006. The spacecraft must leave Earth with a velocity of with respect to the Sun, or with respect to the Earth. The spacecraft will arrive at Jupiter on February 28, 2007 in order to perform a gravity assist. The spacecraft will arrive with a velocity of with respect to the sun, and after a small maneuver to set up the correct hyperbolic trajectory for the gravity assist, will leave Jupiter with a velocity of with respect to the Sun. This will place the New Horizons satellite on a course for Pluto with and excepted arrival date of July 14, 2015. The spacecraft will arrive at Pluto with a velocity of relative to the Sun, or a velocity of relative to Pluto. With the New Horizons spacecraft on an escape trajectory with respect to the Sun, the satellite will now cruise into the Kuiper Belt for the extended mission. Figure 4 shows the trajectory graphically. This plot was created by solving the two-body problem numerically using the Fourth Order Runge Kutta Method with the initial conditions solved for in this analysis. It can be seen that the figure matches Figure 2 very closely.

Figure 4

4. Extension:

The proposed extension to the Pluto New Horizons mission is to visit a second gas giant planet rather than Pluto before venturing off into the Kuiper Belt. Pluto’s status as a planet is in question. While many debate the definition of a planet, Pluto’s small size and highly eccentric and inclined orbit fit in more with the Kuiper Belt than its nearest planetary neighbors, the gas giants. Because of this, many believe that Pluto is simply a large Kuiper Belt object. It can be argued that the reason for going to Pluto is sentimental; as it was discovered and deemed a planet before the Kuiper Belt was known. Meanwhile, the planets Uranus and Neptune have not been explored since the Voyager 2 mission. There have been great advances in technology that can allow observations of these planets that were not possible when the Voyager 2 flew by. Much could be gained from another mission to these planets. For these reasons, this paper proposes that the New Horizons mission perform a close fly-by of Uranus, or Neptune after its gravity assist with Jupiter on its way out to the Kuiper Belt. This would still allow a great deal of scientific observation of the Kuiper Belt, and if Pluto is indeed simply a large well-known Kuiper Belt object, then very little would be lost. However, a great deal of planetary science would be gained, as the spacecraft would perform close fly-bys of two of the planets in the outer solar system.

It is desirable to keep the orientation between Earth and Jupiter the same for the transit between these two planets, as it is already known that this can be done from a fuel and launch vehicle capability standpoint. It is also desirable to keep the path of the spacecraft after leaving Jupiter and heading toward the new planet very similar to the mission already planned for the same reason. In order to determine the most appropriate planet to visit, one must look at the location of the planets at the time of the planned gravity assist with Jupiter. Figure 5 shows the location of each of the planets at the time of the planned Jupiter gravity assist.

Outer Solar System

Figure 5

It can be seen from the Figure that Saturn is not a good candidate for visitation, but Uranus and Neptune do appear to be good candidates. It does not appear that Uranus and Neptune are aligned in a manner that will allow for visitation of both planets before venturing out into the Kuiper belt. If Uranus were trailing Neptune, this would be easier to do, but since it is not, it appears that one or the other must be chosen. In the interest of delaying the launch as little as possible, Neptune will be chosen as the preferred destination. It can be seen from Figure 6 that the planets will be aligned for a fly-by of Neptune before they will be aligned for a fly-by of Uranus. Since the desire is to keep the relationship between Earth and Jupiter the same for the new mission, one must consider the synodic period between the two planets. The synodic period for Earth and Jupiter can be calculated using the following equation:

Assuming that the planets orbits are circular and using the mean motion of the two planets calculated earlier, P=398.87 days. Therefore, in order to maintain the same spatial relationship between Earth and Jupiter, the launch must occur in increments of 398.87 days after the currently planned launch date. It appears as if the ideal launch date for a Neptune fly-by is two synodic periods after the currently planned launch date. This would place the launch date at March 20, 2008. The gravity assist at Jupiter would then occur on May 7, 2009. It can be seen in Figure 6 that on May 7, 2009 the spatial relationship between Jupiter and Neptune is very similar to that of Pluto and Jupiter on February 28, 2007.

Outer Solar System

Figure 6

In order to provide an analysis of the new extended mission, the exact position of Neptune at the desired time needs to be known. In order to determine this, we must choose a desired arrival time. The semi-major axis of Neptune’s orbit is 4,504,449,769 km. Through numerical integration of the transfer between Jupiter and Pluto for the originally planned mission, it can be seen that the spacecraft reached this distance from the Sun roughly 7.429 years after the gravity assist with Jupiter. This will be used as the targeted time for the Neptune fly-by. This would place the Neptune fly-by on October 11, 2016. In order to determine the position of Neptune at this time, the orbital elements must be known. Table 5 shows the orbital elements and the time since periapse at the time of the desired fly-by.

Orbital Elements for Neptune
a	4504449769
e	0.008988095
i	1.76995221°
	131.78405702°
	276.33963348°
Time Since Last Periapse	140.123 years

Table 5

The time since periapse for Jupiter at the time of the gravity assist will be 9.964 years. The time since periapse for Earth at the time of launch will be 75 days. The true anomaly for each of these planets at the desired times can be calculated in the same manner as before. When this is done, one obtains the following:

Now that all of the orbital elements are known for the planets at the critical times, these orbital elements can be converted to position and velocity in the Sun centered inertial system. This can be accomplished in the same manner as before. When these equations are used, one obtains the results shown in Table 6.

	(km)	(km)	(km)	(km/s)	(km/s)	(km/s)
Earth	-149066019.16	1703013.76	0	-0.830	-29.877	0
Jupiter	505849995.26	-566451914.99	-8976224.58	9.591	9.322	-0.253
Neptune	4442698448.71	-579910234.87	-90427786.90	0.664	5.414	-0.127

Table 6

Now that the position and velocity of the planets are known at the critical times, one can use the Universal Variables Solution to Lambert’s Problem just as before to calculate the required initial velocity to reach the planet in the desired amount of time. This problem will be solved for the trek between Jupiter and Neptune first. When this method is used, the following results are obtained:

Again, the Universal Variables Method will be used to calculate the velocity required upon leaving Earth to arrive at Jupiter for the required gravity assist. When this method is used, the following results are obtained:

Just as with the analysis of the planned mission to Pluto, the two sections of the orbit will be pieced together with the gravity assist at Jupiter. In order to analyze the hyperbolic trajectory about Jupiter, the velocity of the spacecraft with respect to Jupiter must be known. The following calculates the velocity of the spacecraft with respect to Jupiter upon arrival, and as it leaves.

It can be seen that in order to reach Neptune in the desired amount of time, it is necessary for . Just as for the planned mission, the radius of periapse will be . The semi-major axis, eccentricity, and angle can be calculated in the same fashion as before to yield the following results.

Again, it can be seen that the velocity upon arrival at Jupiter, and the velocity upon leaving Jupiter do not match up to provide the required hyperbolic trajectory. It is therefore necessary to perform a maneuver upon arriving at Jupiter such that the spacecraft will follow the desired path. For this to happen, the magnitude of the velocity upon arrival must equal the magnitude of the velocity upon departure, the angle between the velocity upon arrival and the velocity upon departure must equal 16.2619°, and the constraint will be imposed that the velocity vector after the maneuver must lie in the same plane as the plane as the initial arrival velocity vector and the planned departure velocity vector. This can be explained with the following mathematical relationship.

When these equations are solved for , one obtains:

The required delta v for the spacecraft to follow the correct trajectory once it reaches the sphere of influence of Jupiter is then:

To summarize the proposed mission to Neptune, the spacecraft would launch from Earth on March 20, 2008. The spacecraft would leave Earth with a velocity of with respect to the Sun, or with respect to the Earth. This would place the spacecraft on a hyperbolic trajectory with respect to the Sun, and place the spacecraft on a course for Jupiter. The spacecraft would be expected to arrive at Jupiter for a gravity assist on May 7, 2009. The spacecraft would arrive at Jupiter with a velocity of with respect to the Sun, or with respect to Jupiter. At this point, in order to place the spacecraft in the correct hyperbolic orbit about Jupiter, a maneuver must be executed. This impulsive maneuver of will give the spacecraft a velocity of with respect to Jupiter. Once this velocity is achieved, the spacecraft will follow a hyperbolic orbit about Jupiter. This orbit will have a radius or periapse of 2359236 km, and a of 16.2619°. The spacecraft will then leave Jupiter with a velocity of with respect to Jupiter, or with respect to the Sun. This will place the spacecraft on a course for Neptune, with a planned arrival date of October 11, 2016. The spacecraft would arrive at Neptune with a velocity of with respect to the Sun, or with respect to Neptune. At this point, the spacecraft would follow a hyperbolic trajectory about Neptune, performing either a gravity assist, or passing in front of the planet and allowing the planet to slow it down. The preferred trajectory would ideally be determined by observations of the Kuiper Belt made prior to the encounter with Neptune. During this phase, the hope is that an interesting Kuiper Belt Object could be targeted, and the correct measures could be taken so that the fly by of Neptune would put the spacecraft on a course to perform a close fly-by of that object. The two-body equation was solved numerically for this trajectory in the same manner as before. The plot of this trajectory is shown in Figure 7.

Figure 7

5. Recommendation:

Based on this analysis, it is my recommendation that the Pluto New Horizons spacecraft go to Neptune on its way out to the Kuiper Belt rather than to Pluto. As was stated previously, Pluto is considered by many to be nothing more than a large Kuiper Belt object that was labeled a planet prior to the discovery of the Kuiper Belt. Because of this, it could be argued that one will learn little or nothing more about the nature of Kuiper Belt objects from visiting Pluto than one would from visiting any other Kuiper Belt object. The proposed trip to Neptune retains the ability to visit the Kuiper Belt, but adds the visitation of a gas giant that has not been visited for quite a long time. There have been a great number of technical advances since the last visitation of Neptune that would allow for additional and more precise observations of the planet that were not possible before. The fly-by of Neptune prior to venturing into the Kuiper Belt seems to offer the most in the way of scientific observation, without compromising other goals of the mission.

Of course, scientific goals are not the only factor contributing to decision-making when it comes to missions of this magnitude. One of the main, if not the most important factor, when considering missions such as these is money. The area where any extension to this type of mission has the highest potential to increase cost is in the required delta v needed for the mission. Increased delta v means increased fuel requirements as well as the potential for a more expensive launch vehicle. The delta v considered for this analysis essentially comes from two sources. The first of these is delta v at launch. The second is the maneuver required to perform the gravity assist. Of course there will be corrective maneuvers here and there throughout the mission, but it is expected that these will be roughly equal for both the currently planned mission as well as the proposed mission and will not be considered. The total delta v breakdown for these two missions is as follows. The delta v required at launch for the currently planned mission is 13.4523 km/sec. The delta v required for the maneuver is 0.8598 km/sec, giving a total delta v for the mission of 14.3121 km/sec. The delta v required at launch for the proposed mission is 11.8692 km/sec. The delta v required for the maneuver at Jupiter is 1.3961 km/sec, giving a total delta v for the mission of 13.2653 km/sec. It can therefore be seen that the required delta v is less for the proposed mission then for the currently planned one. This mission would therefore add no cost due to increased delta v requirement.

Another consideration for this proposed extension to the mission is the schedule. In order for the planets to line up correctly for this mission, the launch has to be delayed by over two years. Arrival at Neptune is less than two years later than the planned arrival at Pluto though, due to the fact that Neptune is closer to the Sun than Pluto at the respective rendezvous locations. While this delay is somewhat of a drawback, it does not appear significant enough to outweigh the other benefits of altering the mission to travel to Neptune rather than to Pluto.

In summary, the benefits of altering the New Horizons Mission to visit Neptune rather than Pluto seem to far outweigh the negatives. While we lose the chance to visit Pluto, we gain the opportunity to perform some scientific studies at Neptune with observational equipment far superior to anything that has gone by Neptune before. We also retain the ability to fly by Jupiter as well as into the Kuiper Belt, while not adding any additional fuel requirements to the mission. Therefore, I believe that this is an opportunity that should not be passed up, and that we should bypass a trip to Pluto in favor of a fly-by of Neptune on the way out to the Kuiper Belt.

6. Conclusions:

The calculations used within this paper are made with a number of assumptions. Among these are the assumptions that the spacecraft is always under the influence of gravity from only one other body, and that the bodies that are providing that gravitational influence can be approximated as point masses. Of course, there are other forces that can act on the spacecraft like perturbations from third bodies, solar radiation pressure, etc. Also, in the case of the gravity assist, the motion of the planet with respect to the Sun during the gravity assist was not considered. The Lambert’s Problem solved for the orbits between the various planets calculated the orbit between the centers of the planets, when of course you want to be offset from the planet a certain degree and not smash right into it. Also, all calculations were made assuming an impulsive delta v. This of course is not really the case, as the velocity will obviously increase steadily during launch. The portion explained in this paper as an impulsive delta v to set up the gravity assist would probably be completed in reality by a series of corrective burns long before arriving at Jupiter. That being said, these types of calculations are very useful for mission planning applications in which one is trying to make initial calculations for the amount of delta v required for a mission and get an idea about the general shape that the orbits will take. While numerical integration will typically be performed in order to determine an exact orbit, the results obtained for the actual Pluto mission are very close to the values quoted in literature. It is therefore believed that the conclusions obtained here are valid, and that the deviations in the results obtained in this paper from real values are small.

This paper was able to successfully apply the concepts of two body dynamics to the problem of interplanetary travel. It was shown that due to the large scale of the solar system, it can be very time consuming to visit the solar systems outermost reaches, even with a very powerful launch vehicle and a gravity assist to help. These same concepts were also successfully applied to a proposed modification to the mission to show the feasibility and requirements of the new mission. Overall, the paper demonstrated the utility of the two-body concepts, and how they can be used to approximate a wide array of problems in the field of astrodynamics.

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