Introduction
Any chosen propulsion system must be capable of successfully transporting the spacecraft from Earth into Jupiter's orbit. Because both planets travel in an elliptical path around the sun, Jupiter's distance from Earth is constantly changing. The distance between Earth and Jupiter varies from 628,743,036 km to 928,081,020 km (from 4.2 AU to 6.2 AU in astronomical units). The interstellar mission will require the spacecraft to travel at least 4.2 AU to the destination. One concern in the mission is the operational lifetime of the equipment on board. Long travel times may lead to deterioration in the performance of the subsystems and various instruments. Thus, it is desirable to develop a trajectory which minimizes the travel time to Jupiter given current technologies. Another concern is how to save fuel during the trip, which directly influences the payload of the rocket and the total cost of this mission. By making a compromise between those two leading factors, the trajectory we choose will employ three gravity assists which will effectively send the spacecraft to Jupiter's orbit.
Launch Vehicle
The primary concerns in the selection of the launch vehicle are the mass of the spacecraft and the desired earth orbit in which the spacecraft will be placed before entering its interstellar trajectory [1]. The desired earth orbit will be a geostationary transfer orbit (GTO). Thus the chosen launch vehicle must have the capability to lift a payload equivalent to the mass of the spacecraft to this orbit. Given that the objective of the interstellar mission will be to receive and transmit information about Jupiter, the systems which primarily contribute to the mass of the spacecraft are the communications and power sub-systems. The estimated mass of the power system required to operate the communication system and transmit signals back to Earth is approximately 3,000kg. Using parametric mass estimating relationship [2], the mass of the spacecraft is expected to be around 19,000kg.
Any chosen propulsion system must be capable of successfully transporting the spacecraft from Earth into Jupiter's orbit. Because both planets travel in an elliptical path around the sun, Jupiter's distance from Earth is constantly changing. The distance between Earth and Jupiter varies from 628,743,036 km to 928,081,020 km (from 4.2 AU to 6.2 AU in astronomical units). The interstellar mission will require the spacecraft to travel at least 4.2 AU to the destination. One concern in the mission is the operational lifetime of the equipment on board. Long travel times may lead to deterioration in the performance of the subsystems and various instruments. Thus, it is desirable to develop a trajectory which minimizes the travel time to Jupiter given current technologies. Another concern is how to save fuel during the trip, which directly influences the payload of the rocket and the total cost of this mission. By making a compromise between those two leading factors, the trajectory we choose will employ three gravity assists which will effectively send the spacecraft to Jupiter's orbit.
Launch Vehicle
The primary concerns in the selection of the launch vehicle are the mass of the spacecraft and the desired earth orbit in which the spacecraft will be placed before entering its interstellar trajectory [1]. The desired earth orbit will be a geostationary transfer orbit (GTO). Thus the chosen launch vehicle must have the capability to lift a payload equivalent to the mass of the spacecraft to this orbit. Given that the objective of the interstellar mission will be to receive and transmit information about Jupiter, the systems which primarily contribute to the mass of the spacecraft are the communications and power sub-systems. The estimated mass of the power system required to operate the communication system and transmit signals back to Earth is approximately 3,000kg. Using parametric mass estimating relationship [2], the mass of the spacecraft is expected to be around 19,000kg.
Based on the payload requirement and cost, Falcon Heavy [3] is selected for this mission. At a listing price of $85M, the Falcon Heavy Rocket is almost $30M cheaper than the Atlas V 551 rocket (used in Juno Mission), the smallest Atlas V rocket that could meet our payload. The Falcon Heavy launch vehicle has the capability of lifting 21,200kg to GTO, or approximately 11% more capability than required. This extra capability will provide a margin for additional mass growth and confirm that the Falcon Heavy will be able to meet our mission requirements. The Falcon Heavy provides a standard fairing as depicted in the image below. At 11.4 m tall with a diameter of 5.2 m, this standard fairing will provide adequate space for our spacecraft. Combinations of acoustic surfaces are used inside the payload fairing to help achieve the acoustic environment. The Falcon Heavy comprises of two stages and bears a similar configuration to the Falcon-9. While the second stage is similar to the Falcon-9, the first stage of the Falcon 9 Heavy consists of two additional Falcon-9 first stages acting as liquid boosters. A single first stage of the Falcon Heavy launch vehicle has nine SpaceX Merlin engines with each engine, powered by liquid oxygen (LOX) and rocket grade kerosene (RP-1), providing 125,000 lbf sea level thrust. Although each mission is unique in its flight profile, a typical flight profile for a maximum-capability geosynchronous transfer orbit mission is shown in the Figure 2 below.
Trajectory
Taking the successful Galileo's VEEGA trajectory design [5] as a reference, the trajectory in this mission will use a set of three gravity assists to reach the destination Jupiter's orbit. The orbital paths of Venus, Earth, Jupiter and the trajectory of the spacecraft are shown on the right. The spacecraft launched from Earth, will undergo a slingshot from Venus and then swing around Earth for another slingshot. After it goes through a full orbit, the Earth will fly by again and provide the last gravity assist sending the spacecraft to Jupiter's orbit. Such trajectory is chosen for a number of reasons. One reason is that it only requires a small amount of fuel for course corrections (delta_v=125 m/s [5]). In addition, the trajectory allows the spacecraft to pass by several asteroids, which provides opportunities for asteroid study. Another reason is that it allows a relatively long launch window. To be more specific, the whole trajectory can be divided into four phases as depicted in the figure below. The first phase will be a Hohmann Type Transfer orbit which will approximately last 112 days for the spacecraft to reach Venus.
Taking the successful Galileo's VEEGA trajectory design [5] as a reference, the trajectory in this mission will use a set of three gravity assists to reach the destination Jupiter's orbit. The orbital paths of Venus, Earth, Jupiter and the trajectory of the spacecraft are shown on the right. The spacecraft launched from Earth, will undergo a slingshot from Venus and then swing around Earth for another slingshot. After it goes through a full orbit, the Earth will fly by again and provide the last gravity assist sending the spacecraft to Jupiter's orbit. Such trajectory is chosen for a number of reasons. One reason is that it only requires a small amount of fuel for course corrections (delta_v=125 m/s [5]). In addition, the trajectory allows the spacecraft to pass by several asteroids, which provides opportunities for asteroid study. Another reason is that it allows a relatively long launch window. To be more specific, the whole trajectory can be divided into four phases as depicted in the figure below. The first phase will be a Hohmann Type Transfer orbit which will approximately last 112 days for the spacecraft to reach Venus.
The Venus encounter will occur on February 10th, 2016. The Venus flyby trajectory is also shown in the above figure. The spacecraft approaches Venus at a closet altitude of 16,123km and the Venus relative velocity is 8.2km/s. [5] After receiving the first slingshot from Venus, the spacecraft will travel another 298 days before it first encounter Earth for the second slingshot. In order for the spacecraft to reach Jupiter from Earth, it must have a heliocentric velocity of about 39km/s. Earth's velocity in its orbit about the Sun is 30km/s, therefore, the spacecraft must acquire about 9km/s in a direction approximately parallel to Earth's velocity vector. The first Earth gravity assist will increase the heliocentric velocity by 5.2km/s, from 30.1km/s to 35.3km/s. This velocity change increases the orbit period from 1 year to 2 years. After completion of this 2-year orbit, the spacecraft will return to the same point for the second Earth slingshot, which will provide the necessary velocity increase for the spacecraft to reach Jupiter. The final orbit period will become 5.6 years.
If we make the following assumptions about this trajectory (which are not necessarily true): (1) planetary orbits are circular and (2) the slingshot is instantaneous in time, which immediately changes the orbit of the spacecraft, an approximate expression of travelling time
can be used for crude calculation. G is the gravitational constant, Msun is the mass of the Sun, Asweep is the swept area of the spacecraft/satellite, Aorbit is the total area of the heliocentric orbit, Rsate is the distance between the start point of the spacecraft/satellite and the Sun, Rplanet is the distance between the encounter point (where the next slingshot happens) and the Sun.
Orbit Transfer
For the mission, the radius of the mothership orbit is 181,000km which results in a 12 hour orbit period. In order to collect information about Jupiter's atmosphere, the CubeSats [6] need to orbit at a lower altitude (radius = 77,000km). Thus, we need to transfer our CubeSats from the initial higher orbit to their target orbit. Basically, the procedure involves imparting a change in velocity on the spacecraft so that it enters an elliptical orbit around Jupiter. This new elliptical orbit will have an apoapsis equal to the radius of the higher orbit, and the its periapsis will be equal to the radius of the lower orbit. It is like a reversed Hohmann Transfer orbital maneuver [7].
Orbit Transfer
For the mission, the radius of the mothership orbit is 181,000km which results in a 12 hour orbit period. In order to collect information about Jupiter's atmosphere, the CubeSats [6] need to orbit at a lower altitude (radius = 77,000km). Thus, we need to transfer our CubeSats from the initial higher orbit to their target orbit. Basically, the procedure involves imparting a change in velocity on the spacecraft so that it enters an elliptical orbit around Jupiter. This new elliptical orbit will have an apoapsis equal to the radius of the higher orbit, and the its periapsis will be equal to the radius of the lower orbit. It is like a reversed Hohmann Transfer orbital maneuver [7].
The equations shown below describe the necessary velocities required to make this transfer happen, where G is the gravitational constant, M is the mass of the primary body (Jupiter), r2 is the radius of the original orbit, r1 is the radius of the target orbit, and (r_1+r_2)/2 is the length of the transfer orbit's semi-major axis.
Because the initial orbit has a radius larger than the ?nal orbit, negative impulses will be required, first at apoapsis and then at periapsis, to decelerate the satellite. Since the length of the transfer orbit's semi-major axis is known to us, the period of the transfer orbit can be calculated by using Kepler's Third Law of Planetary Motion,
Half of the period is the time needed to complete the orbit transfer.
A small rocket will be used to accelerate or decelerate the spacecraft. Based on Newton's Second Law, it is easy to find the relation between the force and the change in velocity:
A small rocket will be used to accelerate or decelerate the spacecraft. Based on Newton's Second Law, it is easy to find the relation between the force and the change in velocity:
Where Fthrust is the force from the thruster, mmax is the mass of the spacecraft before firing, dm/dt is the mass flow rate through the thruster and t is the action time of the thruster.
Below are estimates of the changes in velocity and the forces required by the thruster (assume m = 1,000kg and t = 20s).
Below are estimates of the changes in velocity and the forces required by the thruster (assume m = 1,000kg and t = 20s).
Appendix: Gravity Assist (Slingshot)
In orbital mechanics, a gravitational slingshot is the use of the relative movement (e.g. orbit around the sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed and/or redirect its path. [8]
A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational field of a planet. To simply explain the principle, suppose that you are a "stationary" observer and that you see: a planet moving left at speed U; a spacecraft moving right at speed v. The spacecraft will pass close to the planet, moving at speed U + v relative to the planet. When the spacecraft leaves orbit, it is still moving at U + v relative to the planet but in the opposite direction, to the left; and since the planet is moving left at speed U, the total velocity of the spacecraft relative to you will be the velocity of the moving planet plus the velocity of the spacecraft with respect to the planet. So the velocity will be U + (U + v), that is 2U + v. Of course, a gravity assist follows the conservation of energy and momentum. Because planets are much more massive than the spacecraft, the effects on the planet are negligibly small.
In orbital mechanics, a gravitational slingshot is the use of the relative movement (e.g. orbit around the sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed and/or redirect its path. [8]
A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational field of a planet. To simply explain the principle, suppose that you are a "stationary" observer and that you see: a planet moving left at speed U; a spacecraft moving right at speed v. The spacecraft will pass close to the planet, moving at speed U + v relative to the planet. When the spacecraft leaves orbit, it is still moving at U + v relative to the planet but in the opposite direction, to the left; and since the planet is moving left at speed U, the total velocity of the spacecraft relative to you will be the velocity of the moving planet plus the velocity of the spacecraft with respect to the planet. So the velocity will be U + (U + v), that is 2U + v. Of course, a gravity assist follows the conservation of energy and momentum. Because planets are much more massive than the spacecraft, the effects on the planet are negligibly small.
References:
[1] Timothy Pratt et al., Satellite Communications, 2nd ed. New York: Wiley, 2003.
[2] Wiley J. Larson and James R. Wertz, Space Mission Analysis and Design, 3rd ed. Portland, OR: Microcosm Press 1999.
[3] SpaceX. (2014). Falcon Heavy Available: http://www.spacex.com/falcon_heavy.php
[4] Falcon 9 Launch Vehicle Payload User's Guide, SpaceX, Hawthorne, CA, 2009.
[5] D'Amario, L. A., Bright, L. E., & Wolf, A. A. (1992). Galileo trajectory design. Space science reviews, 60(1-4), 23-78.
[6] Wikipedia. (2014, Jul. 8). CubeSat [Online]. Available: http://en.wikipedia.org/wiki/CubeSat
[7] Braeunig, Robert A. (2012). "Orbital Mechanics," Rocket and Space Technology [Online]. Available: http://www.braeunig.us/space/orbmech.htm
[8] Wikipedia. (2014, Jul. 10). Gravity assist [Online]. Available: http://en.wikipedia.org/wiki/Gravity_assist
[1] Timothy Pratt et al., Satellite Communications, 2nd ed. New York: Wiley, 2003.
[2] Wiley J. Larson and James R. Wertz, Space Mission Analysis and Design, 3rd ed. Portland, OR: Microcosm Press 1999.
[3] SpaceX. (2014). Falcon Heavy Available: http://www.spacex.com/falcon_heavy.php
[4] Falcon 9 Launch Vehicle Payload User's Guide, SpaceX, Hawthorne, CA, 2009.
[5] D'Amario, L. A., Bright, L. E., & Wolf, A. A. (1992). Galileo trajectory design. Space science reviews, 60(1-4), 23-78.
[6] Wikipedia. (2014, Jul. 8). CubeSat [Online]. Available: http://en.wikipedia.org/wiki/CubeSat
[7] Braeunig, Robert A. (2012). "Orbital Mechanics," Rocket and Space Technology [Online]. Available: http://www.braeunig.us/space/orbmech.htm
[8] Wikipedia. (2014, Jul. 10). Gravity assist [Online]. Available: http://en.wikipedia.org/wiki/Gravity_assist