You have a rocket in a high circular orbit around a massive central body (a planet or the Sun) and wish to escape with the fastest possible speed at infinity for a given amount of fuel. In 1929, Hermann Oberth showed that firing two separate impulses (one retrograde and one prograde) can be more effective than a direct transfer that expends all the fuel at once. This is due to the Oberth effect, whereby a small impulse applied at periapsis can produce a large change in the rocket's orbital mechanical energy, without violating energy conservation. In 1959, Theodore Edelbaum showed that this effect could be exploited further by using up to three separate impulses: prograde, retrograde, and then prograde. The use of more than one impulse to escape can produce a final speed even faster than that of a fictional spacecraft that is unaffected by gravity. We compare the three escape strategies in terms of their final speeds attainable, and the time required to reach a given distance from the central body. To do so, in the Appendix we use conservation laws to derive a “radial Kepler equation” for hyperbolic trajectories, which provides a direct relationship between travel time and distance from the central body. The 3-impulse Edelbaum maneuver can be applied to interplanetary transfers, exploration of the outer solar system and beyond, and (in time reverse) efficient arrival and orbital capture. The physics principles employed are appropriate for an undergraduate mechanics course.

## I. INTRODUCTION

Newton's laws of motion and universal gravitation allow us to relate the geometrical
properties of an orbit to conserved quantities such as angular momentum and energy. Changing
the velocity of an orbiting object changes these quantities, and therefore the orbital path.
One of the earliest descriptions of such orbital maneuvers was Newton's cannon,^{1} a thought experiment that demonstrates the
effects of changing a projectile's launch speed on its resulting orbit—increasing this speed
increases the size and period of the orbit. For launch speeds greater than the local escape
speed, the projectile does not return, but follows a hyperbolic escape path and approaches
an asymptotic final speed *v*_{∞} far from the central body.

In the 20th century, pioneers of astrodynamics began to use these principles to chart the
courses of hypothetical spacecraft that propel themselves by the directed expulsion of
matter carried as fuel. The resulting impulses change their orbital paths. Even for a simple
system consisting of a rocket and a massive central body, a variety of maneuvers are
possible, such as Hohmann^{2} and bi-elliptic
transfers between orbits, with differing numbers of impulses employed to use the minimal
amount of fuel.^{3}

In this paper, we discuss three strategies for a spacecraft in a circular orbit to achieve
a high-speed escape with *v*_{∞} > 0. We compare their fuel
requirements to achieve a given *v*_{∞}, and their travel times to
reach a given distance from the central body.

In Sec. II, we start with conservation laws to develop equations that relate orbital speed and distance. In Sec. III, we use these to compare three escape strategies in terms of fuel usage. In Sec. IV, with the aid of the Appendix, we derive expressions for the travel time from the original orbit out to a given distance. In Sec. V, we conclude by discussing applications and providing suggestions for student investigations.

## II. BASIC EQUATIONS FOR ORBITAL MANEUVERS

We assume that the mass of the spacecraft is tiny relative to the central body, such that any change in the latter's motion can be ignored. The central body can then be used to define an inertial frame, in which we analyze the spacecraft's motions.

To simplify our analysis, we consider only impulses of duration much less than the orbital
period around the central body. Equivalently, for such idealized *impulsive* maneuvers the rocket can change its velocity without changing its position, so that the pre-
and post-impulse Keplerian orbits intersect. For such impulses to have the greatest effect
on a spacecraft's orbital energy and angular momentum, they must be applied either prograde
(in the direction of motion, by expelling exhaust backwards) or retrograde relative to the
spacecraft velocity vector. This results in all paths being coplanar, reducing our analysis
to two dimensions. (We discuss orbital plane changes briefly in Sec. V.)

### A. Speed as a function of distance

*M*will have a constant specific energy (where we use “specific” to mean “per unit mass”)

*v*is the magnitude of its velocity vector,

*r*is its distance from the central body's center, and

*G*is the gravitational constant. (This energy properly belongs to the

*system*of the spacecraft and central object, but since the massive central body is assumed stationary, we assign it to the spacecraft for brevity.)

*v*

_{∞}at infinity, an unpowered spacecraft at a distance

*r*

_{dep}must have a departure speed

*v*

_{dep}there given by

*v*

_{A}at

*apoapsis*(farthest from the central body, at distance

*r*

_{A}) and

*v*

_{P}at

*periapsis*(the closest point, distance

*r*

_{P}). Due to the central nature of the gravitational field, the specific angular momentum is conserved and given by

*v*is the azimuthal component of the spacecraft's velocity. Therefore, at the extremes of the ellipse where there is no radial motion

_{θ}*rocket equation*, attributed to Konstantin Tsiolkovsky, which relates the speed change Δ

*v*to the initial and remaining mass of the rocket

*m*

_{i}and

*m*

_{f}are the masses of the rocket (including fuel) before and after the impulse, respectively, and

*v*

_{ex}is the effective exhaust speed.

^{4}(The gravitational attraction between the rocket and exhaust mass can be ignored.) Equation (6) does not depend on the rate of fuel consumption, and applies to multiple intermittent firings of the rocket engine. For this reason, we shall follow the standard convention of using total Δ

*v*as a proxy for fuel cost when comparing strategies to achieve an orbital transfer.

When a mass of fuel (*m*_{i} − *m*_{f}) is
expelled to obtain a given Δ*v*, the chemical energy converted to kinetic
energy is $\Delta E chem= 1 2 m i \u2212 m f v ex 2$. A
fraction of this input energy changes the mechanical energy of the rocket, while the rest
is carried away in the exhaust,^{4} as we
shall discuss in Sec. III E.

## III. ESCAPE FROM A CIRCULAR ORBIT

Initially, we have a spacecraft in a circular orbit of radius *r*_{0} with orbital speed $ v 0= G M / r 0$.
(This relation can be proven by centripetal arguments, or by setting *r*_{A} = *r*_{P} = *r*_{0} in Eq. (5).) We desire to achieve a target
final speed *v*_{∞} for the smallest total Δ*v*, so
that the smallest amount of fuel is required.

### A. Single impulse for direct escape

*v*

_{D}for such a direct escape to produce a final speed

*v*

_{∞}, we use Eq. (2) with

*v*

_{dep}=

*v*

_{0}+ Δ

*v*

_{D}, and

*r*

_{dep}=

*r*

_{0}, i.e.,

This function is plotted as a green dotted line in Fig. 2.

### B. Two-impulse escape—The “Oberth maneuver”

In 1929, Hermann Oberth described a more fuel-efficient strategy^{5} to obtain the same *v*_{∞} from a
high circular orbit (of radius many times that of the central body). First use a
retrograde impulse to drop closer to the body, and then apply a prograde impulse at the
periapsis of that elliptical orbit, as shown in Fig. 1(b). Author Robert Heinlein (who had worked with Oberth on the pioneering
science fiction film *Destination Moon*) described this maneuver in Chapter
7 of his novel *The Rolling Stones*:

“A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward the Earth, then performing her principal acceleration while as close to the Earth as possible.”

^{6}

Reference 7 explains the physics of this *Oberth effect* in detail. If a rocket body of mass *m* moves at speed *v*, a forward specific impulse Δ*v* increases its kinetic energy by $ 1 2m v + \Delta v 2\u2212 1 2m v 2$, which
increases linearly with *v*. Since an orbiting rocket travels faster as it
moves deeper into the potential well, firing an impulse as close as possible to the
central body allows it to maximize the energy gained, in principle without limit as *r*_{P} →0. (In practice, *r*_{P} is
limited by the size of the central body.)

The increase in the mechanical energy of the *system* of rocket+exhaust
comes from the chemical energy converted in the impulse
Δ*E*_{chem}, which is constant regardless of where the impulse was
fired. Therefore, any additional energy gained by a fast-moving rocket must be obtained at
the expense of the mechanical energy carried by the expelled fuel, as we discuss in Sec. III E.

*r*

_{in}of the elliptical orbit produced by the initial, retrograde impulse at radius

*r*

_{0}. From Eq. (5) with

*r*

_{A}=

*r*

_{0}and

*r*

_{P}=

*r*

_{in}, the required velocity change for this first impulse Δ

*v*

_{Ob1}is

*v*

_{P}given in Eq. (5) with

*r*

_{A}=

*r*

_{0}and

*r*

_{P}=

*r*

_{in}. To achieve a final speed

*v*

_{∞}, use Eqs. (2) and (5) with

*r*

_{dep}=

*r*

_{in}to find the second prograde velocity boost Δ

*v*

_{Ob2}from

*v*

_{P}to

*v*

_{dep}

*v*

_{Ob}= |Δ

*v*

_{Ob1}|+ Δ

*v*

_{Ob2}, which simplifies to

*r*

_{in}= 0.05

*r*

_{0}. This curve crosses that described in Eq. (7) at Δ

*v*

_{Ob}= Δ

*v*

_{D}=

*v*

_{0}, independent of

*r*

_{in}, for which the resulting $ v \u221e= 2 v 0$, coincidentally equal to the local escape speed from the original orbit. For desired values of

*v*

_{∞}larger than this, Δ

*v*

_{Ob}< Δ

*v*

_{D}, so the two-impulse Oberth maneuver will require less fuel than the single-impulse direct escape (cf. Fig. 2).

As *r*_{in} is decreased, the “Oberth advantage” (the difference
Δ*v*_{D} − Δ*v*_{Ob}) gets larger for $ v \u221e> 2 v 0$, since the
fuel expelled at periapsis is placed into a lower energy orbit. In the theoretical limit
as *r*_{in} → 0, the dashed purple curve in Fig. 2 becomes flat, and Eq. (8) shows that Δ*v*_{Ob1} →
−*v*_{0} for any *v*_{∞}. This speed
change stops the spacecraft in its original orbit and causes it to fall in radially
towards the center of attraction. Then, the tiniest boost
Δ*v*_{Ob2} at *r*_{in} = 0 could produce
any value of *v*_{∞} desired.

### C. Three-impulse escape—The “Edelbaum maneuver”

In 1959, Theodore Edelbaum described a coplanar escape maneuver that employs three
impulses: a prograde boost to raise apoapsis, followed by a retrograde impulse to cause
the spacecraft to fall in to a low periapsis, and thence a final boost to escape.^{8} An example is shown in Fig. 1(c). By falling in from a larger radius than the
original circular orbit, the resulting faster periapsis speed enhances the Oberth effect,
by transferring more of the expelled fuel's chemical *and* mechanical
energy to the spacecraft.^{7}

*r*

_{A}=

*r*

_{out}, with periapsis at the original orbit radius,

*r*

_{P}=

*r*

_{0}. Equation (5) gives the required post-impulse velocity

*v*

_{P}, from which we find the necessary boost Δ

*v*

_{Ed1},

*r*

_{P}=

*r*

_{in}<

*r*

_{0}. The required velocity change Δ

*v*

_{Ed2}is then

*v*

_{P}of the rocket after it falls from apoapsis at

*r*

_{out}to periapsis at

*r*

_{in}is given in Eq. (5), and the required velocity for escape from there to a specified

*v*

_{∞}is found from Eq. (2) with

*r*

_{dep}=

*r*

_{in}. The difference is the final velocity boost Δ

*v*

_{Ed3}, i.e.,

*v*

_{Ed}= Δ

*v*

_{Ed1}+ |Δ

*v*

_{Ed2}| + Δ

*v*

_{Ed3}, and can be simplified to

*r*

_{in}= 0.05

*r*

_{0}and

*r*

_{out}= 2.5

*r*

_{0}. It is also valid for computing Δ

*v*

_{Ob}if

*r*

_{out}=

*r*

_{0}, and for Δ

*v*

_{D}if

*r*

_{out}=

*r*

_{in}=

*r*

_{0}.

*v*for a given

*v*

_{∞}by an amount

*v*

_{∞}, with corresponding savings in fuel expenditure. (Readers can show that Eq. (15) is positive for any

*r*

_{in}<

*r*

_{0}and

*r*

_{out}>

*r*

_{0}.) This difference can be seen as the constant vertical distance between the dashed purple (Oberth) and dotted-dashed blue (Edelbaum) curves in Fig. 2.

*v*

_{Ed}crosses the dotted green curve for Δ

*v*

_{D}when

*v*

_{∞}, the Edelbaum maneuver requires a lower total Δ

*v*than the direct or Oberth escapes, and so is fuel-optimal for such high-speed escapes.

In the dual limit *r*_{out} → ∞ and *r*_{in} → 0, Δ*v*_{Ed} → $ 2 \u2212 1 v 0$. This
corresponds to a transfer which first sends the spacecraft very far from the central body,
then (much later) applies a very small retrograde impulse to nullify its angular momentum,
causing it to fall radially inwards to *r*_{in} = 0, where a tiny
impulse can produce any value of *v*_{∞} desired. While this *bi-parabolic* trajectory has the smallest possible Δ*v* (≈0.41*v*_{0}) to achieve any desired *v*_{∞}, it requires an infinite travel time.^{9,10}

*v*

_{∞}for an available overall Δ

*v*provided by the fuel,

*r*

_{out}=

*r*

_{0}, and to the direct escape if one sets

*r*

_{out}=

*r*

_{in}=

*r*

_{0}.

### D. Four or more impulses?

A valid question is whether additional fuel savings can be achieved using more than three
impulses. It has been shown that any *fuel-optimal* coplanar transfer will
consist of no more than three impulses.^{10} This also holds true for coplanar bound orbit-to-orbit transfers,
which are either Hohmann (two impulses)^{2} or bi-elliptic (three).^{3} Coplanar escape
strategies that produce fuel-optimal transfers with more than three impulses can always be
reduced to an equivalent with three or fewer.^{11,12}

As an example, consider the following 4-impulse strategy: (1) first slow down from
circular orbit to an intermediate inner radius *r*_{int}, (2) use
the Oberth effect there to boost to *r*_{out}, (3) slow down again
to lower periapsis to *r*_{in}, prior to (4) boosting to escape. We
have shown above that more efficient escapes are realized as *r*_{out} → ∞ and *r*_{in} → 0 (to
increase periapsis speed and hence the Oberth advantage of the final impulse). Therefore,
any intermediate ellipse can be made more fuel-efficient by increasing *r*_{out} and decreasing *r*_{in} to their
mission-constrained extremes. After doing so, the optimal 4-impulse escape degenerates to
the same path as the 3-impulse escape and is no more efficient.

For idealized impulsive maneuvers, a rocket in a closed orbit will always return to the position at which the impulse was applied (unless another impulse is made). A prograde or retrograde impulse at one apsis of an orbit raises or lowers the other apsis. It further follows from Eq. (6) that multiple impulses fired at the same position in orbit are fuel-equivalent to a single, combined impulse.

For example, if the rocket engine cannot apply the retrograde impulse for the Oberth or
Edelbaum maneuver all at once, it can lower its periapsis successively to *r*_{in} by multiple firings each time it passes through
apoapsis, without any loss of overall fuel efficiency. Students can use the equations in
Sec. II to design and evaluate their own
multi-impulse maneuvers.

### E. Energetics of impulsive escape maneuvers

*E*

_{chem}) produce different final rocket speeds

*v*

_{∞}for the three escape strategies? The initial mechanical energy of the rocket+fuel system of mass

*m*

_{i}in a circular orbit is $ E sys , i=\u2212 1 2 m i v 0 2$ from Eq. (1). After all the fuel is expelled, the mechanical energy added to the system is Δ

*E*

_{chem}from Sec. II B, which provides the final rocket mass

*m*

_{f}with a speed change Δ

*v*given in Eq. (6). The final mechanical energy

*E*

_{sys, f}of the system of rocket+exhaust is then

*v*and fuel mass expended, and independent of the number of impulses. Therefore, if one wants the rocket to end up with the highest possible mechanical energy $\epsilon = 1 2 m f v \u221e 2$, one should choose an impulse strategy that leaves the combined exhaust masses with the lowest.

As an exercise, by expressing *v*_{ex} as a multiple of *v*_{0}, students can calculate
Δ*E*_{chem} and the change in kinetic energy of the rocket (due
to changes in its speed *and* mass) after each impulse. The difference
gives the mechanical energy change (positive, negative, or zero) of the fuel mass
expelled. This will reveal for each strategy how the same final system energy *E*_{sys,f} is distributed between the escaping rocket and the
total mass of expelled fuel.

### F. Comparison with a “no gravity” rocket

A remarkable result is that both Oberth and Edelbaum escapes can attain a higher *v*_{∞} than a fictional rocket that is unaffected by the central
body's gravity, which boosts tangentially from orbit at *r*_{0} to
a constant speed *v*_{∞} = *v*_{0} + Δ*v*,
shown as a dotted gray line in Fig. 2. For example,
use Eq. (17) to calculate *v*_{∞} for each escape strategy using a mission total
Δ*v *=* *1.25*v*_{0}. With *r*_{in} = 0.05*r*_{0}, the Oberth escape
gives *v*_{∞}= 2.30*v*_{0}; additionally
setting *r*_{out} = 2.5*r*_{0} gives *v*_{∞} = 2.92*v*_{0} for the Edelbaum
escape—both faster than *v*_{∞} = 2.25*v*_{0} for the “no gravity”
rocket.

The Oberth and Edelbaum trajectories can end up with larger values of *v*_{∞}, because the fuel on board the “no gravity” rocket
carries no gravitational potential energy relative to the central body. With no potential
energy to “steal” from the expelled fuel, that spacecraft's kinetic energy gain is limited
to a fraction of the fuel's chemical and kinetic energy only.^{7}

## IV. TRAVEL TIME COMPARISON

Compared to a single-impulse direct escape, both the Oberth and Edelbaum escapes can
produce faster final speeds for a given total Δ*v* (and thus fuel). However,
these maneuvers require additional elliptical orbit segments where the spacecraft moves
slowly. For very distant destinations (*r* ≫ *r*_{0}),
the extra time on these segments may not be important, but for intermediate distances there
will be a trade-off in total time to destination. In this section, we calculate the travel
times from the initial circular orbit to a specified distance *r* from the
central body, to determine which of the three strategies will reach that distance in the
shortest time.

### A. Hyperbolic segment

All three escape strategies culminate in a hyperbolic escape from a prograde impulse
applied at periapsis. For the direct escape, the departure radius *r*_{dep} = *r*_{0}; for the Oberth and
Edelbaum escapes, *r*_{dep} = *r*_{in}.

*r*from the central body. Equation (A6) gives the time of flight from a periapsis distance

*r*

_{P}out to a destination distance

*r*. We can rewrite that equation in terms of the original circular speed

*v*

_{0}and period

*T*

_{0}= 2π

*r*

_{0}/

*v*

_{0}

### B. Elliptical segments (2- and 3-impulse escapes)

*t*

_{hyp}(

*r*

_{0},

*r*). For the Oberth and Edelbaum transfers, Kepler's third law provides the time spent on the elliptical segments. The transfer time

*t*

_{ell}(

*r*

_{A},

*r*

_{P}) between apoapsis and periapsis is half the period of an orbit of semi-major axis (

*r*

_{A}+

*r*

_{P})/2, as seen in Fig. 1

To obtain the total travel time for the two-impulse Oberth maneuver, we must include the
time for the infalling segment *t*_{ell}(*r*_{0}, *r*_{in}). For the three-impulse Edelbaum maneuver, we must
include two elliptical segment transfer times *t*_{ell}(*r*_{0}, *r*_{out}) + *t*_{ell}(*r*_{out}, *r*_{in}).

### C. Shortest travel time for Edelbaum escape

Expressions for the total travel times for the three escape strategies are summarized in
Table I. These are plotted in Fig. 3 as a function of the Edelbaum maneuver “swing out”
radius *r*_{out}, using the same parameters as for Fig. 2 and a destination distance *r *=* *200*r*_{0}. Clearly for
this case, there is an optimal swing-out radius *r*_{out} ≈
2.5*r*_{0} for the Edelbaum transfer to minimize the travel time.
For larger values of *r*_{out}, the additional travel time in the
first elliptical segment of Fig. 1(c) offsets the
Oberth effect's advantage of gaining a larger *v*_{∞} for the same
overall Δ*v.*

# Impulses | Δv | Total travel time (Eqs. (19)–(21)) |

1 (direct) | Eq. (7) | t_{hyp}(r_{0}, r) |

2 (Oberth) | Eq. (10) | t_{ell}(r_{0}, r_{in}) + t_{hyp}(r_{in}, r) |

3 (Edelbaum) | Eq. (14) | t_{ell}(r_{0}, r_{out}) + t_{ell}(r_{out}, r_{in}) +
t_{hyp}(r_{in}, r) |

1 (no gravity) | v_{∞} − v_{0} | t_{nog}(r_{0}, r) |

There is no tractable closed-form solution for the time-minimizing *r*_{out} as a function of destination distance *r* and mission constraints Δ*v*_{Ed} and *r*_{in}. Instead, students can use Eq. (17), followed by Eqs. (19) and (20), to evaluate travel times for a range of values of *r*_{out} given the values of the other parameters. As the
destination distance *r* increases, the time-optimal value of *r*_{out} also increases, since the elliptical segments take up a
smaller fraction of the overall trip duration, while the larger *v*_{∞} reduces the remaining time spent on the hyperbolic
segment.

*t*

_{nog}for this putative “no gravity” rocket is simply the length of the linear segment from

*r*

_{0}to

*r*divided by its constant

*v*Δ

_{∞}= v_{0}+*v*

## V. DISCUSSION

For a spacecraft that can barely escape the central body (*v*_{∞} ≈
0), Fig. 2 shows that a single-impulse direct transfer
is most efficient in terms of Δ*v*. This is also the case for a spacecraft in
a low orbit that cannot approach the central body any closer than its original orbital
radius *r*_{0}. For a high-speed escape from a high circular orbit,
the Oberth and Edelbaum maneuvers can produce a larger *v*_{∞} for a
given Δ*v*, as shown in Fig. 2. However,
to obtain the benefit of reduced travel times, the destination distance must be much greater
than the original orbital radius, since the Oberth and Edelbaum transfers require slow
elliptical segments prior to the hyperbolic escape. A mission that calls for a high-speed
escape from a lunar-like orbit around Earth^{6} (*r*_{0} ≈ 60 Earth radii and *v*_{0} ≈ 1 km/s), or from a high orbit around any planet or
asteroid, can benefit from employing the Oberth or Edelbaum maneuver.

Most mission concepts are constrained by the total Δ*v* available from the
fuel on board. If employing either the Oberth or Edelbaum transfer, the inner radius *r*_{in} of the intermediate orbit is limited by the radius of the
central body (including its atmosphere), and in the case of the Sun, by heating and
radiation considerations. (The *Parker Solar Probe* makes perihelion passes
as close as 10 solar radii ≈0.05 AU,^{13} but
an escaping spacecraft would only have to do so once.)

To date, no mission has used an Oberth or Edelbaum transfer to send a spacecraft to the
outer solar system. This is mainly because the “Oberth advantage” over direct escape works
best for values of total Δ*v* ≈ *v*_{0} (≈ 30 km/s for
a heliocentric orbit at *r*_{0} = 1 AU), which is not currently
attainable by chemical rockets. Some mission concepts have proposed hybrid
gravity-assist/Oberth trajectories that use a carefully timed fly-by of Jupiter
(*r*_{out} = 5.2 AU) to provide some of the retrograde
Δ*v*_{Ed2} necessary to cause the spacecraft to fall in close to
the Sun.^{14}

Thus far we have restricted our analysis to coplanar orbits and impulses. However, if one
desires to change the plane of the escape hyperbola, the Δ*v* (and fuel cost)
to do so is greatly reduced by applying the plane-change impulse simultaneously with the
second, retrograde impulse of the Edelbaum transfer at *r*_{out},
where the spacecraft is moving slowest.

An important application of the 3-impulse Edelbaum maneuver is in time reverse, for
efficient arrival and capture into a chosen circular orbit when a spacecraft approaches a
planetary body at high relative speed. For approaches with *v*_{∞} > *v*_{0}, the optimal strategy is shown in Fig. 1(c) by reversing the direction of the arrows, and consists
of a braking impulse at periapsis (which could be partially achieved by repeated aerobraking
passes) into an eccentric orbit, followed by a periapsis-raising prograde impulse, and a
third, “circularization burn.”

For student discussions, Table II summarizes pros
and cons of the Edelbaum transfer compared to direct escape and gravity-assist (“slingshot”)
trajectories.^{15} Students can explore how
the functions in Figs. 2 and 3 change as they vary *r*_{in}, *r*_{out}, and total Δ*v* to plan their own escape
and approach maneuvers. They can also extend the analysis presented here to incorporate
noncircular initial orbits, which can alter the choice of fuel-optimal transfer strategy.
Trajectories can be simulated using software such as *Systems Tool Kit* (STK),^{16} NASA's *General Mission
Analysis Tool*,^{17}^{,} *Orbiter*,^{18} or *Kerbal Space Program*.^{19} STK animations that compare the three escape strategies discussed in this paper are provided
in the supplementary material.^{20}

Pros
. | Cons
. |
---|---|

Can maximize v_{∞} for a given
Δv (and fuel expended). In time reverse, provides most efficient
orbital capture. | Only advantageous for high orbits (Fig.1), fast escapes (Fig. 2), and distant destinations (Fig. 3). |

A plane change requires less fuel if executed at r_{out} of the intermediate orbit. | r_{in} is limited by the central
body's size and radiation environment. |

Can be executed at any time (unlike gravity assist). | Requires two engine restarts. |

Mission can be aborted back to the original circular orbit
after the first or second impulse, for a fraction of
Δv_{Ed}. | Spacecraft spends substantial time ∼ T_{0} in the vicinity of the original orbit before
escaping. |

Pros
. | Cons
. |
---|---|

Can maximize v_{∞} for a given
Δv (and fuel expended). In time reverse, provides most efficient
orbital capture. | Only advantageous for high orbits (Fig.1), fast escapes (Fig. 2), and distant destinations (Fig. 3). |

A plane change requires less fuel if executed at r_{out} of the intermediate orbit. | r_{in} is limited by the central
body's size and radiation environment. |

Can be executed at any time (unlike gravity assist). | Requires two engine restarts. |

Mission can be aborted back to the original circular orbit
after the first or second impulse, for a fraction of
Δv_{Ed}. | Spacecraft spends substantial time ∼ T_{0} in the vicinity of the original orbit before
escaping. |

## ACKNOWLEDGMENTS

The authors thank San Diego State University Library staff for assistance with locating aerospace journals published before 1970, an anonymous referee for suggestions, and Donna and Laura Edelbaum for encouraging this work.

### APPENDIX: RELATING POSITION AND TIME FOR A HYPERBOLIC TRAJECTORY

How does an orbiting object's position change with time? Many textbooks derive equations
for this *Kepler problem* using a geometrical method similar to that first
described by Johannes Kepler himself.^{21} This requires the definition of new angular measures (the mean, true, and eccentric
“anomalies”) to locate the object on its conic section path. Such an approach makes sense
for closed periodic orbits, and can be extended to unbound hyperbolic paths.^{22} However, for our purposes we desire a
direct relationship that gives the time taken for an escaping object to reach a given
distance *r* from the central body, starting from periapsis. Here, we
derive such a relationship using only conservation laws that will be familiar to students
of classical mechanics.

#### 1. The radial Kepler equation

*v*and

_{r}*v*, respectively,

_{θ}*v*in terms of the conserved specific angular momentum

_{θ}*h*to form an expression for the radial speed as a function of distance

*r*=

*r*

_{p}, we have

*v*= 0, so from Eq. (A2) we can express the specific angular momentum

_{r}*h*in terms of

*r*

_{p}and

*v*

_{∞}, i.e.,

*h*back into Eq. (A2) to obtain the differential equation

*r*(

*t*) when

*v*

_{∞}> 0, this equation can be integrated numerically to give radial position versus time for an infalling or escaping object.

^{23}Alternatively, to find the time since periapsis as a function of radial distance from the central body, we can integrate the reciprocal of Eq. (A4)

*t*(

*r*

_{p}) ≡ 0. The solution for

*v*

_{∞}> 0 and

*r*≥

*r*

_{P}can be written as

*h*=

*0 and hence*

*r*

_{p}= 0.

^{24}For comparison with traditional derivations,

^{22}the

*hyperbolic anomaly*is given by the inverse hyperbolic cosine term in Eq. (A6).

#### 2. Plotting the hyperbolic escape trajectory

*r*

_{p}> 0, the spacecraft's angular position

*θ*can be described in polar coordinates in terms of

*r*and the conic section's semi-major axis

*a*and eccentricity

*e*

*θ*= 0 at the periapsis distance

*r*

_{P}=

*a*(1−

*e*); the bounds on

*θ*constrain the object's position to the correct branch of the hyperbola between its asymptotes. The geometrical parameters

*a*and

*e*are obtained from

*v*

_{∞}and

*r*

_{p}using the standard results

^{22}

*a*<

*0 and*

*e*>

*1. Examples of such paths are shown in Fig. 1.*

## REFERENCES

*A Treatise of the System of the World*

*The Attainability of Heavenly Bodies*

*v*

_{ex}such that the mechanical energy input per unit fuel mass expelled is given by $\n \n 1\n \n 2\n \n v\n \n e\n x\n \n 2$.

*Ways to Spaceflight*