In 1928 Oberth[1] suggested the option of a two-burn orbital maneuver that would, on the first burn, drop an orbiting spacecraft further down into the central body’s gravity well, then a second burn would be performed to accelerate the spacecraft allowing it to escape the gravity well.  This maneuver was not studied in detail by Oberth.  Later, a brief analysis was done by Levin[2].  A sketch of the maneuver is illustrated in Figure 1. 

Fig 1. In the two-burn maneuver, the first burn decelerates the spacecraft resulting in a trajectory that takes it further into the gravity well of the central mass (this is opposed to a single direct burn which accelerates the spacecraft). The second burn is performed at the periapse of the new elliptical orbit and accelerates the spacecraft out of the gravity well of the central mass. 

In two-body motion the summation of forces on the vehicle is simply: [math]!ddot{vec{r}}=frac{mu}{r^2}left(frac{vec{r}}{r}right)[/math] Where: [math]!mu=GM[/math] From the above equation the gravitational force is a function of the inverse square of this radius vector.  In the Newtonian equation only the mass of the large body is considered since the mass of the vehicle is much smaller.  Taking the dot product of equation (1) with respect to  [math]vec{r}[/math] yields: [math]!xi=frac{V^2}{2}-frac{mu}{r}==frac{mu}{2a}[/math] where  [math]xi[/math] is referred to as the specific orbital energy of the vehicle.  The orbital energy is the sum of the specific kinetic energy (the first term) and the specific potential energy (the second term).  In deriving this equation it is convenient to assume that the reference line for potential energy is at infinity.  The right hand side of the equation expresses specific orbital energy as a function of the orbits semi-major axis.     The equations above define the major parameters of a spacecraft in a coasting trajectory around a central body.  In the operation of a spacecraft, changes in the orbit must be made to allow the spacecraft to travel to points of interest such as other planetary bodies.  Turning on the spacecraft propulsion system, or performing a ‘burn’ in industry parlance, will place the vehicle in a new coasting trajectory once the burn is completed.    Due to the computationally intensive nature of trajectory design, there has been strong interest since the 1950’s to determine analytical approximations to low thrust trajectories that would give generally accurate results.  A prevalent argument in the derivation of these approximations is that the thrust vector should always be aligned with the spacecraft’s velocity vector.  The reason can be seen by taking the time derivative of the kinetic energy equation, which is: [math]!frac{d}{dt}KE=frac{d}{dt}frac{V^2}{2}=vec(V)dotvec{a}[/math] Thus the instantaneous rate of change of kinetic energy is proportional to both acceleration and velocity.  The local maximum is found when the spacecraft velocity and acceleration are parallel.  However, as argued by Levin[3], the spacecraft could accelerate in a different direction, forcing the spacecraft into a different orbit with a point of closest approach to the central body, the periapse, being closer to the central body than the original orbit.  Examination of equation (2) shows that for a coasting orbit, the specific mechanical energy remains constant but the kinetic energy is traded for potential energy.  At periapse kinetic energy is at a maximum and potential energy is at minimum, just like for any gravitational force dominated problem like a swinging pendulum or a ball in free flight.  By equation (3), driving to a lower orbit and then accelerating at periapse would maximize the change in energy of the spacecraft for a given acceleration (and thrust).  The optimal method for increasing V to increase a would be to decelerate, coast to a lower orbit where V is maximized (at periapse) and accelerate again.  It has been shown that this maneuver will produce more specific orbital energy than a direct burn out when the total to be produced exceeds the initial circular velocity around the central body.  In equation form: [math]! Delta V_1+Delta V_3[/math] which is always greater than: [math]!sqrt{frac{mu}{r_0}}[/math] The discussed two-burn option is easily confused with a gravity assist maneuver. However, the gravity assist maneuver is based on a massive body such as a planet dragging the spacecraft along for part of the spacecraft’s trajectory.  Momentum (and specific orbital energy) will be exchanged between the planet and spacecraft.  The effect on the spacecraft is substantial, imparting in most cases a velocity change [math]left(Delta Vright)[/math]  that could not be easily duplicated with current propulsion systems.  The effect on the planet is minimal, due to its massive nature relative to the spacecraft.  This momentum exchange is between an external body and the spacecraft and the exchange will occur even if no burn is made by the spacecraft.  Conversely, a slingshot maneuver will not work unless there is a substantial burn at the periapse of the elliptical trajectory.  The additional energy gained by the spacecraft is represented by the additional loss in specific energy by the propellant expended at the periapse burn.  It does not represent a transfer of momentum from the central body to the spacecraft.     The two-burn option can produce a greater specific mechanical energy for a given [math]left(Delta Vright)[/math] budget than a direct burn but only when the total budget exceeds the initial velocity in the initial orbit.  So the [math]left(Delta Vright)[/math] budget must be considerable before the slingshot maneuver is worthwhile.  For instance starting from a circular orbit around the sun at a distance equal to earth’s orbit, the [math]left(Delta Vright)[/math] budget equal to the initial circular velocity is sufficient to completely escape the solar system with a  [math]V_infty[/math] of about 17.5 km/sec.   However the  [math]left(Delta Vright)[/math] budget is well within the range of many missions of interest to NASA. For instance the interstellar precursor mission presents the challenge of traveling 1000 astronomical units (AU) within 50 years, the career lifetime of the average engineer or scientist.  The escape velocity above will deliver a spacecraft to the required distance in over 110 years so clearly a slingshot maneuver would be useful for this mission.  Other deep space missions to the outer planets, Kuiper Belt, Oort Cloud, and heliopause would similarly be enhanced by use of this maneuver.                 Finally a class of mission that has received attention by NASA in recent years is the deflection or fragmentation of asteroids and comets that are on a collision course with Earth.  The [math]left(Delta Vright)[/math] imparted to an oncoming asteroid is very low, on the order of 1-100 cm/sec[4].  This  [math]left(Delta Vright)[/math] is sufficient to deflect most asteroids provided that the impulse is applied to the asteroid early enough.  Current deflection methods require 2-50 years between application of the impulse and the projected collision date.  Therefore the device that will impart the impulse to the asteroid must intercept or rendezvous with the asteroid with all haste.  Given the above the [math]left(Delta Vright)[/math] requirement to intercept an incoming asteroid is generally on the order of 10-30 km/s[5].  The [math]left(Delta Vright)[/math] requirement to rendezvous can be as high as 70 km/sec.  Both values are well within the range necessary to make the two-burn maneuver economical.                 This paper started by examining the concept that acceleration along the velocity vector would result in an optimal acceleration of the spacecraft.  While acceleration along the velocity vector is locally optimal it turns out that there is a special maneuver that in certain cases will outperform the “optimal” cross product acceleration by actually decelerating the vehicle and accelerating it when it reaches periapse.  It is presented here as an important maneuver to be considered for high  [math]Delta V[/math] missions such as interstellar precursor or similar deep space missions and potentially crewed round trip missions to Mars and beyond.  This has profound implications for future space exploration.  Being able to use in-situ resources to create propellants or even construct vehicles on the moon has even greater importance now that the two-burn maneuver can be used to substantially reduce the propulsive requirements for deep space missions. The  required [math]Delta V[/math] to complete a mission to Mars and return is 12-13 km/sec for Hohmann transfers.  However, radiation exposure, crew supplies and crew mental health issues have forced vehicle designers to look at much higher [math]Delta V[/math] missions that can reduce trip times and mission risk.  Mission studies for crewed missions to Mars with a limiting total trip time of 2 years of less have  [math]Delta V[/math] requirements above 20 km/sec.  Therefore it is possible that the new option could have application in orbit raising maneuvers where limiting mission time is critical.  A proof of principle calculation shows significant gains in performance for crewed missions to Mars using this maneuver.  This gain is predicated on the hope that water ice will be found on the moon and can successfully be turned into useable propellant.  Given this assumption the two-burn maneuver can reduce vehicle size by up to half, or decrease mission time by half.  The former dramatically reduces the cost of a Mars trip, while the latter reduces the risk to crew.                  The two-burn maneuver shows considerable promise to enable a variety of scientific and exploration missions in deep space.  The authors believe that this two-burn maneuver could have as large of an impact on space exploration as the gravity assist.  Developed at the very beginning of the space program, the gravity assist enabled missions from Voyager to Cassini to visit the planets of the solar system using technologies that were then available.  Clearly without the gravity assist those technologies would have been inadequate to explore much of the solar system outside of the moon and Mars.  Similarly the two-burn maneuver provides a method for exploration of the boundary of the solar system and interstellar space using today’s technologies and technologies of the near future.  Such missions are difficult to conceive without considering the advantages of alternative maneuvers. References: [1] Oberth, Hermann, Ways to Spaceflight, translated from German in NASA TT F-622. [2] Levin, D. F.., “Escape to Infinity from Circular Orbits”,Journal of the British Interplanetary Society, Vol. 12, No. 2, March 1953, pp.68-71. [3] Levin, E., “Low Acceleration Transfer Orbits”, Section 9.1 in Handbook of Astronautical Engineering, edited by Heinz Hermann Koelle, McGraw-Hill book Company, New York, 1961. [4] Near-Earth Object Survey and Deflection Analysis of Alternatives, Report to Congress, NASA, March, 2007. [5] Adams R. B., Alexander, R., Bonometti, J., Chapman, j., Fincher, S., Hopkins, R., Kalkstein, M., Polsgrove, T., Statham, G., White, S., Survey of Technologies Relevant to Defense from Near-Earth Objects, NASA, TP-2004-213089.