This concept applies to either arrivals or departures from a
planetary body. Arrival and departure
would take place at the periapsis of an extended elliptic orbit about the
planet, where the orbit speed is closest
to any required arrival or departure spacecraft speed, all measured with respect to the planet. The min and max radii of the ellipse
determine both its shape and its speed distribution.
Figure 1 shows everything you need to compute from
min and max distance any of the elliptic orbit parameters, given a mass and radius for the central
body, and a value for the universal
gravitation constant. Figure 2
shows a numerical example computed for a nominal 4-day extended ellipse about
the Earth, with a perigee altitude of
300 km, to match “typical” low circular
Earth orbit conditions.
From low circular orbit,
the ideal dV required to get onto the co-planar ellipse from circular
orbit would be the ellipse perigee speed minus low circular orbit speed. Any interplanetary trajectory has a
“near-the-Earth” speed requirement that is always beyond escape speed, sometimes significantly. To get onto that interplanetary trajectory
from the extended ellipse, the ideal dV
is trajectory speed minus perigee speed,
a smaller number. To get onto
that same trajectory directly from low circular orbit, the ideal dV would be near-Earth trajectory
speed minus circular orbit speed, which
is a much larger number! Same applies to
arrivals, just reversed in direction.
To get a perigee speed that is close to escape at perigee
altitude, the ellipse has to be quite
extended. Its apogee distance will
generally be well outside the geosynchronous distance, and it will usually fall outside the outer
Van Allen radiation belt as well. But
such an extended ellipse will transit the two major Van Allen belts twice each
passage, both outbound, and both inbound. (Note that the geosynchronous distance
actually does fall within the outer Van Allen belt.)
Figure 1 – All the “How-To” for Elliptic Capture Orbits
Anywhere
Figure 2 – An Elliptic Capture Orbit of 4 Day Period About
the Earth
These dV values are ideal in the sense that they are
astronomically-derived values. Depending
upon whether the propulsion is “impulsive” or not, these may need to factored-up higher for the
mass ratio-effective values necessary to use the rocket equation for sizing
stages or vehicles. This allows only for
gravitational losses, since there are no
drag losses outside of the atmosphere.
If the propulsion is “impulsive”,
a factor f = 1 may be used. For
electric propulsion as we currently know it,
use a factor f between 1.5 and 2.
I prefer the more conservative 2,
some others recommend only 1.5.
The value I use for the universal gravitation constant is G
= 6.6732 x 10-11 N-m2/kg2. Values for planetary body masses M and radii
are given in the table just below.
The orbit equations work in N, kg,
m, and sec units of measure. However,
it is customary to show distances in km not m, and speeds in km/s not m/s. The values for G and planet M “go
together”, in the sense that it is their
combined effect that determines orbit characteristics. Do not mix values from different sources!
Figuring interplanetary speed requirements is out-of-scope
here. Suffice it to say that differences
Vfar between body and spacecraft with respect to the sun must be
corrected for 3-body attraction as distances close, which is then the Vnear needed at
departure. Vnear2
= Vfar2 + Vesc2, escape figured at periapsis, not the surface. The magnitudes are accurate, but there is no direction information!
More Detail
Note that elliptic capture (or departure) takes place at the
extended elliptic orbit’s periapsis,
where speed is the highest and closest to arrival and departure speed
requirements. It does not take
place at the ellipse’s apoapsis, where
speeds with respect to the planet are lowest,
and very far indeed from interplanetary arrival or departure speeds (as
measured with respect to the planet).
Note also that for the extended ellipse, the periapsis speed is very close to the
escape speed at that altitude. That
means there is a large dV required to get onto the ellipse from circular
orbit, very nearly the difference
between escape speed and circular orbit speed (usually near 30% of
escape). This also applies going back to
circular from the ellipse, just reversed
in direction. To do this, the burns must take place at periapsis, not apoapsis.
That is part of the fundamentals of making changes to orbits: a burn changing
the speed at one end, changes the
distance at the other end.
Now, there is one
additional nuance of using an extended elliptical capture orbit, and that would be to put its periapsis
altitude down in the atmosphere,
somewhere between the entry interface altitude (140 km for Earth) and
the surface (0 km altitude). The idea
would be to use an aerobraking periapsis pass to decelerate from interplanetary
arrival speed (near the planet) to the calculated periapsis speed for the
ellipse penetrating the atmosphere.
Instead of departing hyperbolically,
the vehicle then follows an extended ellipse out of the atmosphere, without making any burn at all. See Figure 3.
If nothing else is done,
the vehicle will inevitably enter the atmosphere, upon returning to periapsis passage, which is still within the atmosphere. If instead,
the orbit needs to be modified to move that periapsis above the
atmosphere, that is done with an
apoapsis burn (of rather modest magnitude).
But the move to a low circular orbit will still require a periapsis burn
of rather large magnitude (roughly 30% of escape), no matter what. See Figure 4.
This one aerobraking pass into an extended ellipse that gets
modified into a stable orbit with a small apoapsis burn, is an attractive capture method, excepting the difficulty of reaching this
orbit from either the surface or low orbit,
because the dV to reach it terminates very nearly in escape speed, not just low orbit speed! That difference is always inherently on the
order of 30% of escape at the periapsis altitude. If the vehicle so capturing must also contain
the propellant to change to a low orbit,
this attractiveness entirely evaporates! It is actually “cheaper” by the value of the
stabilizing apoapsis burn, to decelerate
directly into low orbit from the interplanetary trajectory, always outside the atmosphere (and thereby
eliminating the need for a heat shield).
One way around this dilemma is as follows: after the initial hard aerobraking pass
(which always requires a deep penetration, or you will NOT get the
deceleration!!!), you adjust periapsis
with a small apoapsis burn, such that
the periapsis altitude just barely falls below the entry interface
altitude. That way, over repeated passes, the drag deceleration at periapsis reduces
apoapsis altitude on each pass,
ultimately toward the desired final circular orbit value. Then one small burn at that final desired
apoapsis altitude pulls your periapsis up out of the atmosphere, for a stable low circular orbit. See Figure 5.
Implicit in this process is the requirement to decelerate
hard on the first aerobrake pass, deep
in the atmosphere. This is imposed
by a required deceleration dV on the order of a nontrivial fraction of
escape, or else you will not capture at
all! The subsequent
apoapsis-lowering decelerations can be much smaller, not nearly so deep in the atmosphere. Gentler is more such orbital passes, over a longer period of time, though. It’s a tradeoff.
The hard braking required on that first pass deep in the
atmosphere implies that the vehicle must have a very significant heat shield! You do not get such deceleration amounts
unless you go deep in the atmosphere: there
is no “shallow-skimming” that gets you any significant deceleration! And if you get deceleration, you WILL get heating! The peak heating pulse precedes the peak
deceleration pulse in all entries.
Further, if the
vehicle uses multiple shallow passes to adjust apoapsis after the initial deep
deceleration pass, its heat shield
must remain serviceable for multiple entries, each a little less demanding than the
preceding one. The initial one occurs at
near-escape speed at periapsis. As each
subsequent shallower pass decreases the apoapsis, periapsis speed decreases toward circular
orbit speed, which is still quite the
demanding entry in terms of heating (at Earth).
Finally, in Earth’s
atmosphere, which has repeatable and
predictable density vs altitude at entry altitudes, this multi-pass aerobraking elliptic capture
process has much merit.
But on Mars, where
the high-altitude densities vary through factors of plus-or-minus two (or
more), rather unpredictably, this aerobraking deceleration process has far
less merit! As you leave the
atmosphere on the first pass, if your
speed is still too high, you will have
to burn to decelerate, lest you not
capture at all, becoming quite literally
“lost in space”! (Mars entry
interface is 135 km, Earth’s is 140 km.)
If you have to be prepared to do that, you might as well avoid the uncertainty
and risk, and just decelerate with a
burn directly into some orbit, all
outside the atmosphere. And not need
any heat shield!
Figure 3 – Aerobraking Elliptic Capture With No Burn, Resulting in a Second-Pass Entry
Figure 4 – Aerobraking Elliptic Capture, With A Burn To Raise Periapsis Out Of the
Atmosphere
Figure 5 – Aerobraking Elliptic Capture With A Burn To Allow
Subsequent Braking Passes
Yet More Detail
The aerobraking capture notion (into an extended capture
ellipse) is conceptually illustrated in Figure 6. The simple 2-D Cartesian entry spreadsheet
model that I have, cannot do this analysis, although a typical entry plot is shown in
that figure for a simple direct entry off Hohmann at Mars, with a big heavy object, which did use the simple spreadsheet analysis. The point of including it was to show
that peak heating is seen before significant deceleration gees are seen! And that will always be true, in any entry!
In addition, at the
figure’s bottom, a couple of sketches
are provided to conceptually show what an actual aerobraking capture event must
entail. All the dV to capture into
the extended ellipse must be obtained in the first pass, by definition! Or else it is not capture at all!
The entering object has a significant dV requirement in
order to capture, which must be produced
by aerobraking drag, by definition in
this scenario. To accomplish that
requires that the entering object experience significant deceleration
gees, although perhaps not actually the
peak possible value in a straight entry.
To obtain those significant levels of deceleration in only the 1
pass, it must inherently dive
rather deep into the atmosphere,
to somewhere near (or even deeper than) where peak convective heating
will occur. (If at Earth or Venus where
speeds exceed 10 km/s at entry interface,
there will also be even-larger amounts of plasma radiation
heating.)
If the required deceleration dV is not achieved in the
first pass, the object is literally lost
in space, likely exiting at a speed
still above escape, unless it can
quickly burn to make up the difference. Bear in mind that the extended-ellipse periapsis
speed is only very slightly below escape speed at that entry interface
altitude, while the hyperbolic
approach speed off Hohmann transfer will be significantly above escape speed
at that same altitude. For faster
transfers, the approach speeds are, in point of fact, well above escape, easily by as much as 50-60% of escape!
Figure 6 – Summary of the Aerobraking Capture Process Into
an Extended Ellipse
Conclusions
1. (1) Elliptic capture makes sense at Earth where
high-altitude densities are reliably predictable. The “right” process is that in Figure 5, where one deep pass captures into the ellipse, with an apogee-raising of the perigee to a
higher altitude, but still in the
atmosphere, where the apogee altitude
can be reduced by drag deceleration at perigee in multiple circuits.
2. (2) Elliptic capture does not make sense at
Mars, where high-altitude densities can
vary up or down by a factor of 2 or more,
erratically and unpredictably, in
terms of current known science. If you
must be prepared to burn to make up an unpredictably-deficient first pass
deceleration, you might as well just
burn directly into orbit, all outside
the atmosphere, entirely eliminating the
need for any heat shield. The dV is
about the same either way.
3. (3) There is no “shallow skimming” of the
upper atmosphere to decelerate in one pass into elliptic capture, without the need for a full entry-capability heat
shield. You either decelerate
significantly for capture (and suffer heating) during the initial pass, or you escape into “lost in space”
status, if you cannot burn to make up
the deceleration deficit. The very
significant heating occurs before you ever see any significant deceleration
gees, and you must see
significant deceleration gees, in order
to capture at all, per Figure 6.
4. (4) Further:
to use repeated shallow passes to reduce elliptic capture apogee per
Figure 5, your full entry-capability
heat shield must be capable of surviving repeated heating episodes, although only the first pass is the
worst. Materials embrittled upon cooling
after the first (deep) pass will fall apart under the influence of pressure
forces and re-heating effects, even
during shallow subsequent passes.
Other Final Comments
1. (1) Entries off the interplanetary trajectories at
Mars (from Earth) take place somewhere in the 5 to 8 km/s range, depending upon the speed of the
interplanetary trajectory. Only convective
heating is significant in this speed range,
which varies roughly proportional to speed at entry interface cubed. The plasma sheath is more-or-less transparent
to infrared, so that refractory heat
shields that re-radiate to cool are feasible,
just as they are at Earth from low Earth orbit, at about 8 km/s at entry interface (this is not
true of entry from extended ellipses about the Earth).
2. (2) Entries at Earth and Venus off of interplanetary
trajectories take place at speeds in the 12-17 km/s range, for which plasma radiation heating dominates
over convective by far, varying as some
very high exponent equal to or exceeding 6,
of speed at entry interface.
Under these conditions, the
plasma sheath is not at all transparent to infrared re-radiation, so that only ablatives are feasible, according to all known technologies.
3. (3) The new heat shield concepts based around carbon
fabrics on spars (resembling umbrellas),
or around carbon or other materials extended as inflatables, are NOT reusable! Per the designs and the testing, they are only good for one heat exposure! There is shrinkage, cracking,
and embrittlement, in all of
those materials, after only one heating
exposure. Re-heating on a subsequent
exposure means a re-exposure to the wind forces, whereupon the embrittled or shrinkage-cracked
materials will fall apart!
