Elliptic Capture

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 V_far between body and spacecraft with respect to the sun must be corrected for 3-body attraction as distances close,  which is then the V_near needed at departure.  V_near² = V_far² + V_esc²,  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!