Thursday, October 18, 2018

Taking a Knee Is Way Over-Politicized


I see too much influence of sometimes-vicious political propaganda regarding football players protesting by taking a knee (or locking arms) during the national anthem.  That propaganda says they are disrespecting the flag and the nation by not standing;  I disagree.

I don’t see any of that supposed disrespect.  None of these players are talking or laughing,  unlike many in the stands!  Their attention is quietly focused on the anthem ceremony,  just as it should be.  Only their posture is “wrong”,  and that is the attention-getting item that makes the protest successful. 

Protest is indeed as American as apple pie.  This nation was quite literally born out of protest!  The most famous of these protests was the Boston Tea Party,  but there were many over several years,  before the shooting started that became the American Revolution. 

The protest issue itself aside,  what’s different here is the utter lack of any real collateral damage!  Most protests as we have known them lead to people getting hurt or property damage being done.  Nobody is hurt,  and no damage is done,  when these players kneel! 

I applaud that;  it’s the most beneficially innovative form of protest I have ever seen.  Of course,  it’s television showing it,  that makes it as effective as it is. 

As for the protest issue,  the statistics verify that something is indeed wrong with equality-of-justice in this country.  It’s probably a lot more complicated than any one explanation,  things usually are.  

But if there’s a problem,  simple ethics demands we investigate and try to correct it. 

The protest exists because we haven’t been paying attention to this problem.

--------------------------------

To support my point,  here are two photos taken during the anthem ceremony at a couple of football games.  The first is fan behavior,  which is largely pretty good.  Most are standing,  some are singing. 

I see some crossed arms (usually symbolic of disapproval of,  or impatience with,  what is going on),  a few talking instead of singing or paying attention,  and at least one individual paying attention to a cell phone instead of the ceremony (dead center,  about 3/4 of the way down the photograph). 

The second is of players,  which are kneeling in protest.  All are quiet,  and all are focused on the ceremony.  Other than the “wrong” posture,  where is the disrespect?  No one is laughing or talking,  no one is focused on anything but the ceremony. 


Fans



Players

Friday, October 5, 2018

Space Radiation Risks: GCR vs SFE

I got my data on risks and passive shielding from NASA’s own site and documents.  See

http://srag.jsc.nasa.gov/Publications/TM104782/techmemo.htm, titled Spaceflight Radiation Health Program at JSC (no cited reference newer than 1992).

or go to http://exrocketman.blogspot.com and click on year 2012 then May 2,  for the article titled “Space Travel Radiation Risks”.  That article,  which I wrote and posted,  abstracts the most relevant information from the NASA source for the questions at hand. 

Two Risks:

There is galactic cosmic radiation (GCR) and there are solar flare events (SFE).  GCR is a very sparse trickle of extremely high-energy particles (mostly protons) that are very hard to passively shield,  and which can induce secondary showers of other dangerous particles in some materials.  The risk is modulated by the solar wind which varies with the solar activity level.

SFE radiation is mostly proton and heavier particles emitted from the sun sporadically,  from eruptions on its surface.  These come in very concentrated but brief events,  comprising much lower-energy particles that are far easier to passively shield. 

GCR:  Roentgen equivalent man (REM) per year  = 42 + 18 sin(360o (t, year)/ (11 year)),  max 60 at solar min,  min 24 at solar max.  Solar max is max sunspot activity,  with more frequent eruptions.  Although,  eruptions can occur throughout the cycle.

SFE: from 1968 to 1970,  events every month or so ranging from 2 REM to 50 REM accumulated during each event;  from 1970 to 1972,  events about every 6 months ranging from about 50 REM to about 100 REM accumulated during each event;  and in 1972 right between Apollo 16 and Apollo 17,  one event right at 5000 accumulated REM.  There was one event of about 5 REM during Apollo 16.  Figure 1 is a plot of SFE events during the Apollo program,  direct from the NASA document’s Figure 10.  The quoted numbers are for somebody outside a spacecraft wearing only a spacesuit,  per NASA’s figure.

By way of comparison,  the Earthly natural background in the US is near 300 milli-REM (0.30 REM) per year. Worldwide is not significantly different. 

A rough rule of thumb:  500 REM accumulated in a short time (hours or days) is considered lethal to 100% of those so exposed. 

NASA’s Astronaut Exposure Rules:

These vary with the affected organ (eyes,  skin,  and 5 cm inside the body as representative of blood-forming organs or “BFO”),  but the lowest values are for 5 cm inside.  NASA’s exposure rules limit exposure to 50 REM accumulated in any one year,  25 REM in any single month,  and a career limit that varies with age and gender,  but peaks at 400 REM accumulated over an entire career. 


These are illustrated with Tables 1 and 2 lifted from the NASA document,  and presented here as Figures 2 and 3,  respectively.  Use the equations in Figure 3 to calculate career limits.  These exposure limitations represent approximately twice the exposures nuclear workers are allowed to face,  with a single-digit percentage increase in cancer risk expected. 

 Figure 1 – SFE Events During Apollo,  from the NASA Document’s Figure 10

 Figure 2 – NASA Time Interval Exposure Limits

 Figure 3 – NASA Career Exposure Limits (Use the Equations)


Effectiveness of Shielding Materials:

These divide into the effects of aluminum,  water,  and hydrogen,  which fairly well bounds a lot of possible materials.  The true risk here is the SFE event, of a very large magnitude,  such as the 5000 REM 1972 event.  That’s outside in nothing but a space suit.  Inside the command module,  the effect of the spacecraft structure reduces the exposure to 500 REM.  That’s a 10:1 reduction for the spacecraft hull (remember,  SFE particles are lower-energy and far easier to shield passively).

Based on the NASA document’s Figure 9,  given here as Figure 4,  the plot for the 1972 event reduces 500 REM inside the command module to 20 REM at 20 g/cm2 of aluminum shielding added to the effects of the spacecraft hull.  Use the density of aluminum (2.7 g/cm3) to find the actual physical thickness of this aluminum to be 7.4 cm.  You would want an aluminum shield that thick or thicker to survive a 1972-magnitude SFE event,  and stay barely within the month exposure limit.

The effect of the various materials as passive shielding for GCR is given in Figure 5,  which is lifted from the NASA document’s Figure 6.  In terms of shield mass per unit area of hull,  hydrogen is the most effective,  and aluminum the least,  with water in between.  Note how the curves flatten at larger masses per unit area,  leading to stronger differences in the amount of shield material required.

Here we ignore the shielding effect of the spacecraft hull structure as negligible against the more energetic radiation.  For 20 g/cm2 (7.4 cm thick) aluminum,  60 REM/year reduces to 40 REM/year,  which is well within the annual limit.  You get the same exposure at only 10 g/cm2 water at 1 g/cm3,  which is 10 cm thick,  and 3 g/cm2 hydrogen,  which at .07 g/cm3 is some 43 cm thick. 

The problem with “over-killing” the shielding is the secondary shower of dangerous particles created by the high-energy GCR particles.  This gets to be a problem if you make the shield too thick,  and it simply doesn’t happen with the lower-energy SFE particles.  That makes passive shielding a real trade-off for design purposes.  This is less a problem with spacecraft design,  and more of a problem with surface habitation design and construction.  The temptation would be to pile too much regolith atop the roof,  causing the secondary scatter problem.  

 Figure 4 – Effectiveness of Aluminum Against SFE Events,  from NASA Document Figure 9

Figure 5 – Effectiveness of 3 Materials Against GCR,  From NASA Document Figure 6


My own recommendation would be to use 15-20 g/cm2 water,  some 15-20 cm thick,  which would reduce 60 REM/year GCR to about 30 REM/year.  Against the GCR,  the same protection obtains at some 50 g/cm2 aluminum,  which is about 18.5 cm thick.  These are similar thicknesses (15-20 cm water vs 18-19 cm aluminum),  but very different masses per unit area:  15-20 g/cm2 water vs 50 g/cm2 aluminum.  Water is simply the lighter shield for the same effect, by about a factor of 2.5 to 3.   

We don’t have anything but aluminum to look at in Figure 4 the for the SFE event.  At the same 50 g/cm2 aluminum that looks good against GCR,  the 500 REM SFE event,  as measured inside the command module,  gets reduced to a 2 REM/event exposure.  Assuming the same attenuation ratio and mass per unit area ratio between aluminum and water that we saw with GCR in Figure 5,  then we should see the same 500 REM to 2 REM reduction of SFE radiation that 50 g/cm2 of aluminum provides,  with only 15-20 g/cm2 water,  which is 15-20 cm thick.  That’s an assumption requiring verification

Lessons for Spacecraft Design:

Water is the best shielding material,  because it is the lightest,  while providing practical thicknesses,  unlike hydrogen.  My best guess is that storable propellants should resemble water in their shielding properties.  They are light molecules made of light atoms,  like water,  and have densities far more comparable to water than to liquid hydrogen.

The recommended water shield is 15-20 g/cm2 (15-20 cm thickness),  which could be water,  wastewater,  or even frozen food.  It could also be storable propellants like the hydrazines and NTO oxidizer.  Increase the thicknesses for wastewater,  ice,  or frozen food:  a good guess is a factor of 1.5 to 2 increase in thickness over straight water.

Against GCR and ignoring any effects of the spacecraft hull,  15-20 cm of water should reduce 60 REM/year of GCR to something near 30 REM/year.  This might,  or might not,  be practical for the entire spacecraft,  but putting water in one form or another around the sleeping quarters might be,  in addition to a designated radiation shelter space.  

There is some benefit of the spacecraft hull reducing SFE from a 5000 REM/event outside the hull to a 500 REM/event inside the hull.  Since that is still a lethal dose,  further shielding is absolutely required!  That same 15-20 cm of water should reduce an inside-the-hull 500 REM/event to something nearer 2 REM/event,  which is well within the monthly limit,  even if multiple such events occur spaced rapidly together.  This gives us a lot of margin in the case of an event far larger than the 1972 SFE event.

Any spacecraft design should incorporate its flight control station within the designated-shelter radiation shielding so that critical maneuvers may be flown regardless of the solar weather.  Shielding about the sleeping quarters is also recommended for purposes of reducing GCR exposure. 

By using the shadow-shield effect,  this kind of shielding might be obtained with a combination of water/wastewater/frozen food items located about the sleeping quarters and flight control station,  combined with propellant tanks for the next burn,  that are docked about the periphery of these regions,  outside the spacecraft hull.  These are all items you already must have,  anyway,  so that extra shielding mass is not added to the design.  See Figure 6 for a concept sketch.

 Figure 6 – Concept for Incorporating Propellants and Water-Based Materials as a Shadow-Shield


Exposures Calculated for a Mars Mission at 60 REM/Year GCR with Three 1972-Class SFE Events:

The mission is 9-month transit/13-month near (or on) Mars/9-month transit.  One SFE event occurs during each transit,  and the other occurs while the crew is on or near Mars.  Calculations are made with and without the sleeping quarters shielded,  for 1/3 of clock time during each day.  Shielding about the sleeping quarters and the designated shelter is spacecraft hull plus 15-20 cm water-equivalent for SFE,  just 15-20 cm water-equivalent for GCR.  How this might actually be done was shown conceptually in Figure 6 above.

The first radiation exposure year is 3 months pre-mission on Earth,  then 9 months in transit to Mars.  The second radiation-exposure year is 12 months on Mars.  The third radiation-exposure year is one month on Mars,  9 months in transit to Earth,  and 2 months post-mission on Earth.  Earthly exposure is at the 0.3 REM/year rate.

The 9 month transit is 0.75 year.  Without any shielding effects at all,  45 REM are accumulated during transit for the year in which transit occurs. 

If there is sleeping quarters shielding,  its presence cuts the GCR to a rate of 30 REM/year while sleeping. Then based on clock times,  a 2/3-1/3 split occurs between the 60 and 30 REM rates:  that is a rate of 50 REM/year applied to a 9 month transit.  Thus the crew accumulates 37.5 REM during the transit,  which goes toward the total accumulated exposure during the year in which the transit takes place.

While on or near Mars,  the planet blocks half the spherical “sky”,  for a net in-space unshielded GCR rate,  assumed unattenuated by the planet’s atmosphere,  of 30 REM/year,  accumulated during each year spent on Mars.  The stay is 13 months,  so without sleeping quarters shielding,  30 REM counts toward the first full year on Mars,  and one month’s worth (2.5 REM) counts toward the second year on Mars and in-transit home.   

If there is shielding about the sleeping quarters on Mars,  the same 2/3-1/3 split applies to rates of 30 and 15 REM/year,  reducing the effective exposure rate to 25 REM/year.  In that case,  the year on Mars accumulates 25 REM,  and the 13th month accumulates 2.1 REM. 

Add 2 REM to the accumulation in any one month for SFE events during the transits and during the stay on Mars.  There must be a designated radiation shelter for SFE events,  even while on Mars,  or the exposures could easily be lethal.  That assumes no attenuation of the radiation by Mars’s thin atmosphere. 

The result is depicted graphically in Figure 7.  Again,  shielding is 15-20 cm of water-equivalent.  


Figure 7 – Radiation Profiles for 15-20 cm Water Shield,  with or without Sleeping Quarters Shield


These results show a marginal yearly exposure during year 3,  at just barely under the 50 REM annual limit,  for the case of no sleeping quarters shielding.  With sleeping quarters shielding,  this reduces well under the limit.  Worst case monthly exposures are well under the limit for both cases.  Two such missions will approach career limits,  in either case. 

Note that if there is no shielded place for SFE events,  then during any event in the same class as the 1972 event,  the exposure will be fatal at 500 REM received over a matter of hours.    There simply must be a solar flare shelter somewhere.  This is true during the transits and on Mars.

Note also that during times when the GCR is under 60 REM/year out in space,  exposures inside the ship (either case) will be much lower.  It is only the worst-case 60 REM/year space environment that is analyzed here.

Note also that it is the shorter-than-a-year transit time that reduces in-transit unshielded exposure to 45 accumulated REM!  Extending the transit time by using repeated aerobraking passes to capture at Mars,  instead of a one-time rocket burn,  will quickly violate the annual exposure limit!  The same thing applies to electric propulsion using spiral-out/spiral-in flight plans.  (Not to mention exposure times passing through the Van Allen Belts at Earth.)

Once the unshielded one-way flight time exceeds 10 months,  the annual exposure limit gets exceeded in a max GCR year.  Once that long-transit situation obtains,  you must shield the entire habitable volume of the ship,  not just a designated shelter and perhaps the sleeping quarters. 

Finally,  if the entire habitable volume of the spacecraft could be shielded at the 15-20 cm water-equivalent level,  exposures would be cut essentially in half,  to only around 30-something REM per radiation-exposure year in the transits,  and 15 REM/year on Mars,  even in a 60 REM/year part of the solar cycle.  Year 1 would be 32.08 REM,  year 2 would be 17 REM,  and year 3 would be 33.3 REM,  for a mission total of 83.4 REM.  Three,  or possibly even 4,  such missions in 60 REM GCR years might be feasible,  before hitting career exposure limits.  

Final Comments:

First,  this kind of radiation shielding will inevitably prove to be absolutely necessary,  but it will look nothing at all like what we have ever before done with our spacecraft designs.  When “they” show you spacecraft design concepts that look like what we have done before,  you already know that ”they” have not thought this problem through!

Second,  the shield design concept shown in Figure 6 above is entirely compatible with a “long” ship design that is spun end-over-end like a rigid baton for artificial gravity.  A design like that is also entirely unlike anything we have ever before done,  but it is rather well-understood from an engineering viewpoint,  and would require far less technology development and demonstration than any sort of cable-connected spin gravity design.

Third,  it is quite evident that worst-case GCR risks are slight over-exposure for late-in-life cancer,  while the SFE risks really are lethal doses leading to an ugly death within hours.  Thus,  when “they” point to GCR as the radiation risk that precludes humans going into deep space,  you already know that (1) they are lying for nothing but fear-mongering purposes,  and (2) “they” are truly ignorant of the real radiation risk.  Such claims are simply not credible.

Wednesday, October 3, 2018

On the Senate Kavanaugh Accuser Hearings


Cases like this are supposed to be settled in a proper court of law,  not the "court" of public opinion.  Before the 24/7 news "cycle",  the media were not so insistent on trying these cases in the "court" of public opinion. 

The "court" of public opinion is not a cousin,  but a sibling,  of lynch mob justice!  That has a very poor track record at determining true outcomes,  which is why we supposedly outlawed it (except that the media and our political figures have brought it back). 

And don't kid yourself,  an open Senate hearing is nothing but another “court” of public opinion.

The proper way of dealing with this would have been to do the police investigation,  and present the results to a grand jury.  If worthy of an indictment,  then go to trial with it.  If not,  safely discard the issue.

Meanwhile,  if the accused is a public figure being considered as a nominee for a judicial post,  then (1) he is innocent until proven guilty,  but (2) you don't want to have to unseat him for a conviction,  afterwards,  if he does go to trial.  It is difficult to unseat judges,  even with proven misbehavior.

What that really means is that you put the nomination on hold,  until you find out from the real courts of law whether he really is innocent or guilty.  I'm sorry,  that is the wisest choice,  and it is best for the good of the country.  Simple common sense says so.  Very inconvenient for party advantage,  though.

We seem to have so very few who prioritize the good of the country above party (or personal) advantage anymore.  My advice is vote only for those who would prioritize highest the public good,  regardless of their party membership.  All else pales in comparison.

GW


Monday, September 24, 2018

Relevant Data for 2018 BFS Second Stage

I did the very best I could,  reverse-engineering what the 2018 version of the BFS second stage might be able to do.  This is based on the Musk presentation of a paying moon passenger,  at Spacex,  and posted on their website in Sept. 2018.   I had to use the 2017 rocket engine data at 250 bar chamber pressures,  as the 300 bar performance figures for 2018 are not yet available.  I did not revisit the first stage BFR (see ref. 1).  Things begin with a best-cut guess at the 2018 weight statement: 

All metric ton                     2017                      2018
Payload                                150                         100
Inerts                                    85                           135
Burnout                               235                         235
Propellants                         1100                       1100
Ignition                               1335                       1335

Note:  payload reduced by 50 tons while inerts are increased by that same 50 tons,  due to fins and the longer payload section.  The 3 fins are heavier than the old 4 landing legs,  but inherently span wider.

                                             2017                       2018
Mass ratio                           5.6809                   5.6809
Propellant fraction           0.82397                 0.82397
Payload fraction                0.11236                 0.07491
Inert fraction                      0.06367                 0.10112
Sum fraction                      1.00                        1.00

About the Engines:

                                             2017                       2018
Engines                                6                              7
SL                                        2                              7(nom.) to 0
Vac                                       4                              0(nom.) to 7

                                             2017                       2018
Pc, bar                                  250                         300
Throttle, %                          20-100                   xxx
SL Fth @ SL, KN                 1700                       xxx
SL Isp @ SL, s                      330                         xxx
SL Isp @vac, s                      356                         xxx
SL De, m                               1.3                          xxx
Vac Fth @ vac, KN              1900                       xxx
Vac Isp, s                              375                         xxx (380?)
Vac De, m                            2.4                          xxx

“Xxx” means actual data hinted at,  but not available yet

BFS weights more-or-less determine SL-vac mix of engines:

Scenario               M, ton  1-g KN   .384-g KN
Landing                235         2305       885
Takeoff                1335       13,092    5027

To land on Earth with SL engine design (250 bar):  use 3 engines at 45+%,  if 1 lost,  remaining 2 at 68+%

To land on Mars with vac engine design (250 bar):  use 2 engines at 23+%,  if 1 lost,  remaining 1 at 47+%

To land on Mars with SL engine design (250 bar):  use 2 engines at 26+%,  if 1 lost, remaining 1 at 52+%

To take off fully loaded on Earth with SL design (250 bar):  use all 7 engines at 110% (fully loaded takeoff not feasible)

To take off fully loaded on Mars with vac design (250 bar):  Use 4 engines at 66+%,  if 1 lost,  remaining 3 at 88+%

To take off fully loaded on Mars with SL design (250 bar):  Use 4 engines at 74+%,  if 1 lost,  remaining 3 at 99+%

Engine-Mix Conclusions (for 250 bar designs):

#1. BFS used only at Earth could use all SL engines,  or use 3 SL engines to land,  and 4 vac engines for better Isp to LEO.  Requires BFR first stage with 31 SL engines.

#2. BFS used at Mars and returning to Earth must use 3 sea level engines for Earth landing,  and 4 vac engines for best takeoff from Mars,  as well as powering to LEO.  Requires BFR first stage with 31 SL engines.

For Figuring Performance:

3 SL engines for Earth landing:  Isp = 330 s,  Vex ~ 3.236 km/s

4 Vac engines for Mars landing,  or for powering to LEO:  Isp = 375 s,  Vex ~ 3.677 km/s

Test flight BFS-only takeoff 7 SL engines:  Isp = 330 s,  Vex ~ 3.236 km/s,  max TO mass (to hover only) 1213 metric tons vs 1335 tons fully loaded

Estimating BFS Performance:

These are jigger-factored rocket equation estimates,  per the methods of ref. 2.  The orbital mechanics delta vee requirements come from ref. 3.  For landing at Mars,  retro-burn starts near end of hypersonics at very low altitude,  near 0.7 km/s flight velocity (see landing estimates below).  For landing on Earth,  the “skydiver” descent rate at low altitude appears from Musk’s presentation to be ~ 0.2 km/s.  For getting to LEO from the stage point from BFR,  a slightly-factored delta-vee is orbit velocity minus stage velocity.  Staging velocity is presumed to be ~ 3 km/s. 

Powering to LEO on 4 Vac engines (250 bar design):

Stage velocity 3 km/s,  orbit velocity 7.9 km/s,  theo. dV = 4.9 km/s.  Apply 5% grav-drag loss:  dV = 5.1 km/s.  Req’d MR = exp(5.1/3.677) = 4.00;  Wp/Wig = 1 – 1/MR = 0.75,  vs 0.82 available (8.5% margin).  Margin is 3% at 2.5 km/s staging velocity.  Therefore,  the presumption of 3 km/s staging velocity,  or perhaps slightly lower,  is thus verified. 

Departing LEO and landing upon Mars,  using 4 vac engines (250 bar design):

Depart LEO dV = 3.9 km/s,  land on Mars dV = 1.0 km/s (factored from 0.7 km/s by 1.4),  total = 4.9 km/s.

Req’d MR = exp(4.9/3.677) = 3.791;  Wp/Wig = 1 – 1/MR = 0.736,  add 10% for boiloff to 0.810,  with only 0.823 available (1.5% margin implies,  at full payload,  Hohmann min energy transfer only!!!!).

Departing Mars on 4 vac engines,  and landing upon Earth on 3 SL engines (250 bar design):

Earth free fall = theo. min dV to land = 0.2 km/s,  factor by 1.5 to 0.3 km/s;  req’d MR = exp(0.3/3.236) = 1.0971 (figured from SL perf.);  dWp/Wig = 1 – 1/MR = 0.089;  add 10% for boiloff:  dWp/Wig = 0.098.

Loaded Mars takeoff on 4 vac engines direct to min energy Hohmann interplanetary trajectory:  min theo. dV = 5.35 km/s,  factor up 2% for gravity and drag,  dV = 5.46 km/s;  req’d MR = exp(5.35/3.677) = 4.284 (figured for vac perf.);  dWp/Wig = 1 – 1/MR = 0.767;  total Wp/Wig = 0.865,  with only 0.823 available at full rated payload!  Therefore,  payload must reduce!

Estimate takeoff Wig = 1100 tons propellants/.865 = 1272 tons.  The difference 1335-1272 = 63 tons is the required payload reduction for the return trip,  with no propellant margin at all.  Max return payload = 100 – 63 = 37 tons,  and that is for a min-energy Hohmann transfer trip!!! 

Miscellaneous Information:

There is not much change,  if any,  to the 31-engine first stage (BFR).  The real changes are a lengthened payload section and 3 large fins,  for the second stage (BFS).  The vertical fin is fixed (and termed more of a landing leg than a fin by Musk),  while the other two articulate about hinge lines for aerodynamic control during entry and landing.  These 3 fins replace the four folding landing legs previously shown

The articulation varies from roughly 45 degrees away from the vertical fin during entry and descent,  to a 120 degree separation at landing,  and during initial boost at launch.  Per Musk,  actuation forces for the articulated fins are “in the mega-Newton class”.  See Figure 1 and Figure 2.

The best-estimated landing sequences are shown in Figure 3.  Musk’s September 2018 presentation included a landing computer simulation video that he showed twice.  It was clearly an Earth entry and landing,  as effective deceleration to subsonic in the vertical-descent “skydiver” broadside-to-the-wind mode,  would be impossible to achieve in the thin air on Mars. 

For the Mars landing,  the 2017 presentation’s computer simulation video is still the best guide,  leading to a very low-altitude transonic pitch-up into a sort of tail-slide maneuver,  to position the vehicle tail-first for its final touchdown.  However,  it is likely that thrust must be used to effect the pitch-up into the tail-slide,  because lift equal to weight requires Mach 2-to-3 speed in such thin air.  

That means landing thrust must start at end-of-hypersonics at about Mach 3 (about 0.7 km/s).

For those worried about the fin tips digging into the soil on Mars,  here are some allowable soil bearing pressure data for selected Earth materials,  which might be similar to some soils on Mars.  Design practice requires static exerted pressures be less than these allowables.  For dynamic events,  design practice says stay under half these allowables.  The ton in the data is the 2000 lb US ton.

Ton/sq.ft             MPa                       type

1-2                          0.1-0.2                  fine loose sand

4-6                          0.38-0.58              compact sand and gravel,  requiring picking

8-10                       0.76-0.96              hardpan,  cemented sand and gravel,  difficult to pick

10-15                     0.96-1.43              sound shale or other medium rock,  requiring blasting to remove

25-100                   2.4-9.56                solid ledge of hard rock,  such as granite,  trap,  etc.

Eyeballed Fin Dimensions, Etc.:

Looking at the BFS images in Figures 1 and 2,  we might estimate fin dimension root-to-tip as about equal to basic body diameter,  which is said to still be 9 m.  That puts the fin tips about 13.5 m off of vehicle centerline.  With articulation to 120 degree spacing,  these tips form an equilateral triangle as the “footprint”. 

That puts the shortest distance from the vehicle centerline to the adjacent footprint edge (halfway between two tips) at about 6.75 m.  The “span” from there to the opposite fin tip is 6.75+13.5 = 20.25 m.  The vehicle itself is over 50 m long,  so the height to effective span ratio is about 2.5 to 3.  For the 2017 design with 4 landing legs,  this fell in the 3-4 range.  Some slight improvement in rough-field landing stability may have been obtained,  by going to the fin-as-landing-leg approach. 

The rounded tips on the rear tips of the fins cannot be more than 1 m diameter,  as eyeballed from the images.  That puts the total supporting bearing area for 3 fins at about 2.35 sq.m.  Exerted static bearing pressure at landing weight on Earth is 0.98 Mpa,  and on Mars is 0.38 MPa.  Exerted static bearing pressure at BFS-only takeoff weight on Earth is 5.6 MPa,  and 2.1 MPa on Mars. 

Mars regolith in many places looks like sand and gravel requiring picking,  in other places like loose fine sand.  It would appear the BFS could land on the sand and gravel requiring picking,  but not the loose sand.  However,  it cannot take off from that sand and gravel,  because the weight after refilling with propellant requires a medium rock to support it without sinking-in,  and getting stuck,  or possibly toppling over and exploding.  Prepared hard-paved pads appear to be fundamentally necessary for this design,  unless the fin tip landing pad area can be at least tripled.

Issues Not Fully Explored Here,  But Still Quite Critical:

#1. Rough field landings:  both soil bearing pressures and overturn stability on rough ground or because of obstacles under a landing pad. This requires serious attention!!!

#2. How to seal organic-binder carbon composite propellant tank structures against propellant leakage,  and also have this sealing (and the basic structures) survive at cryogenic temperatures.  None of this has been made public yet. 

#3. How to keep hot slipstream gases from scrubbing the leeside windows and composite structure.  These hot scrubbing flows result from the flow fields at high angle-of-attack,  that are induced by vortices shed from the strong body crossflow component,  and from the nose-mounted canard tips.  See sketch in Figure 4!  This can be a very serious issue for window failure.  It was for the Space Shuttle.

#4. How much internal pressurization is required to resist broadside airloads during entry and descent?

#5. No designs have yet been presented for cargo and tanker versions.  In particular,  the tanker design affects how many tanker refilling flights are necessary for BFS to depart from LEO. 

#6. Estimated costs per launch from Spacex are unavailable.  Some things seen recently on the internet suggest ~ $300 million per launch.  For 100-ton payloads,  that is ~ $3 million per ton,  for the one flight.  Such figures are entirely unreliable as yet,  and likely will remain so,  until several flights into LEO have been made. 

References:

#1. Article dated 4-17-2018 and titled “Reverse-Engineering the 2017 Version of the Spacex BFR” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.

#2. Article dated 8-23-2018 and titled “Back-of-the-Envelope Rocket Propulsion Analysis” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.

#3. Article dated 9-11-2018 and titled “Velocity Requirements for Mars” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.




 Figure 1 – BFS/BFR at Launch,  2018 Version

 Figure 2 – BFS/BFR at Staging,  2018 Version

 Figure 3 – Best-Estimate Analysis of BFS Earth and Mars Entry and Landing,  2018 Version


Figure 4 – How Crossflow Vortices Greatly Enhance Lee-Side Heating Rates

UPDATE 9-28-18:  The shortage of fin tip bearing area can be addressed fairly-easily by a relatively minor shape change as indicated in Figure 5.  Instead of a tip pod with a round landing pad,  make the tip installation a larger part of the fin tip,  with an elongate pad.  Figure 5 shows the bearing area comparison between three 1-m dia round pads,  and three elongate pads 3.6 m x 1 m overall. 

This reduces the fully-loaded takeoff bearing pressure on Mars from 2.1 MPa to 0.49 MPa.  That reduction falls within the safe range for desert hardpan,  and might even be allowable for some simple compacted sand and gravels (requiring picking).  Landing (lighter vehicle weight) at 0.087 MPa becomes no problem for these types of soils on Mars,  even simple loose sand.  Although,  that loose sand is still quite unacceptable for supporting refilled takeoff weight.

Being able to land and take off from loose Mars sand is governed by takeoff weight (5027 KN),  and requires a total bearing area of about 50.3 sq.m to stay under 0.1 MPa bearing pressure.  That is probably far outside what is geometrically feasible. 


Therefore,  the unimproved landing sites are restricted to compacted sand and gravel requiring picking,  or better,  even with the elongate pads shown here.  




Figure 5 – How to Increase Landing Pad Area In the Simplest Way


UPDATE 10-1-18:  A few astute individuals have expressed a concern about BFS landing pads exposed to hypersonic heating,  if built as a streamlined item a the rear of the fin tips,  or as a part of fin trailing edges,  as I proposed just above. 

The best shape for a landing pad is not known to me,  but it is unlikely to be anything like a streamlined shape.  Better to design it to support the weight of a BFS fully-fueled on Earth (some 13,092 KN),  for purposes of short-hop flights.  It seems likely this is a relatively flat-surfaced shape,  whether round in footprint,  or elongated,  as advocated just above. 

This pad is also very likely to be of substantial weight,  bearing as it does the full Earth weight of a fully-fueled BFS,  with due allowance for impact effects during the landing transient.  It is also very likely to a surface that is hydraulically extended,  with shock-absorbing partial retraction,  much as any shock absorber.  And it is very likely to need the bending strength to endure hogging and sagging over obstructions,  instead of uniform pressure. 

If you retract the heavy pad itself just inside the otherwise wide-open fin trailing edge,  it sees no hypersonic scrubbing action,  only simple subsonic wake turbulence,  albeit at a high temperature.  Given the short duration of the entry event,  and the weight of a substantial structure,  the pad needs no heat protection to heat-sink its way through the entry event. 

This situation is sketched in Figure 7.  Bear in mind that the original Mercury and Gemini capsules had metal surfaces in contact with the hypersonic wake.  These were thin but structurally-unloaded corrugated skin panel structures capable of considerable radiative cooling,  surviving quite well at 8 km/s entry speeds from Earth orbit.  Using the old rule-of-thumb,  that’s around 8000 K gas temperatures,  with considerable ionization into plasma.  Free-entry/above-escape entry interface speeds are in the 6-7 km/s range at Mars (about 6000-7000 K),  and 11-17 km/s at Earth (about 11,000-17,000 K,  with very considerable radiation heating from the plasma sheath at all speeds above 10 km/s). 

The landing pad structure should probably be cellular,  in order to have lots of bending strength,  while not allowing any significant debris accumulation.  This is also shown in Figure 7.  If one pad can support the entire weight of the ship,  for obstructions touching anywhere on the pad undersurface,  then we have factor-3 redundancy to cover transient impact loads during landing. 

Looking at load,  shear,  and moment diagrams for the rock-under-the-middle and rock-under-the-end cases,  we find the same max moment magnitude to resist,  just opposite signs.  That value under these assumptions is 5.9 MN-m = 52 E6 inch-lb.  For the lateral dimension of 0.5 m,  the moment arm and moment magnitude are less than for the rock-under-the-end case.  Base the cellular spacing on 5.9 MN-m,  and the pad will be strong enough. 

For a typical high-alloy hardened steel,  yield strength might fall in the 50,000 to 100,000 psi range.  Use 75,000 psi,  and find the necessary pad section modulus (for the long direction) S ~ 700 in3.  Ignoring the section modulus effects of the top and bottom surfaces,  and just considering rectangular-section verticals 15 inches tall and half an inch thick,  the section modulus per vertical is 141 in3.  We only need about 5 such verticals spanning a meter-wide enclosure,  so the spacing is about 7.8 inches = 3 cm.  That’s 85% open volume in a square grid. 

Make the bottom out of the same kind of half-inch alloy steel plate,  and the top out of sheet metal.  That’s about 4000 lb = 1800 kg each for a 3.6 x 1 m pad footprint.  Very small for the 3 pads (5.4 metric tons) compared to the mass of the vehicle (~ 1300 metric tons). 

My conclusion is that there is really no reason why this cannot be made to work safely and reliably.  It will take very careful detailed structural-thermal design,  more than what I did here.  The max soak-out temperatures need to fall below the annealing temperature of the selected alloy,  so that properties do not change with age and number of flights. 


Looking at the reactions in the figure,  each hydraulic cylinder (of a pair per elongated pad) should be capable of providing the Earth weight of the BFS vehicle as an extension force.  That impacts the design of the landing pad hydraulics,  something beyond scope here.  


Figure 7 – Landing Pad Rough-Cut Design Data


Update 10-3-18:  For those who want to see how I calculated performance numbers,  see Figure 8.  This is simple rocket equation work,  with the required kinematic delta-vees jigger-factored upwards to account for gravity and drag losses,  or for severe uncertainties landing.  I used factor 1 for in-space departure from LEO,  factor 1.02 for the gravity and drag-affected departure from Mars,  and factor 1.5 upon the touchdown burn delta-vees. 

The “kicker” that throws off the simplest calculation is the 10% evaporation or boiloff loss for cryogenic propellants during the 9 month transit to Mars.  Propellant remaining after the departure burn is knocked-down 10% in the weight statement to start the arrival sequence. 

Another “kicker” is the change in specific impulse for the Earth landing with sea level engines.   The vacuum bell design cannot be used for that.

What I get doing it this more realistic way is a propellant-remaining safety margin upon landing that is a single-digit percentage of the original propellant load at departure.  It corresponds to approximately 1 km/s extra speed from the departure burn,  without really affecting the landing.  This does eliminate all the safety margin for obstacle avoidance or correcting trajectory errors during the arrival.  It is something I would not recommend!

I was surprised and pleased to find that these performances were not so very sensitive to the actual payload carried.  Raising the payload to Mars from 100 tons to 150 tons cut the 9% margin to 5%. Raising the payload back to Earth from 37 tons to 50 tons cut the 7% margin to 6%.   


Reducing propellant load from 1100 tons to 900 tons cut the to-Mars margin from 9% to 5%,  and the Earth-return margin from 7% to 4%.  These margins are thus demonstrably more sensitive to the initial propellant load carried.  The lesson is:  always top off the tanks completely,  before you fly.  


Figure 8 – Some Details for BFS Performance Estimation

Tuesday, September 11, 2018

Velocity Requirements for Mars

One starts with the interplanetary trajectory from Earth to Mars.  That can be a min-energy Hohmann transfer orbit,  or something more energetic.  The more energetic trajectories require more than by-hand estimates,  and also require more propellant expenditures and vector addition,  so only the min-energy Hohmann transfer orbits are covered here. 


There is not one single min-energy orbit,  because the length of its major axis is the sum of distances of Earth and Mars from the sun.  These distances vary,  because the planetary orbits are eccentric, more so Mars.   You have to look for worst case arrival and departure conditions,  and design for those,  so you can fly anytime.  The effects of this are shown in Figure 1.  Posigrade is counterclockwise,  on this figure. 

 Figure 1 – Min-Energy Hohmann Transfer Orbits to Mars

The figure shows planetary orbital velocities at aphelion,  average,  and perihelion distances,  for both Mars and Earth.  For the bounding cases of Mars aphelion/Earth aphelion,  Mars perihelion/Earth perihelion,  and the average-distances case,  transfer orbit aphelion and perihelion velocities are shown. 

Velocities of the vehicle Vinf when it is “far” from Mars and Earth are also shown.  These are just the difference between the planet’s orbital velocity and the transfer orbit velocity,  in effect a coordinate change from sun-centered to planet-centered.  One-way travel time (half the orbital period) is also shown.

The aphelion/aphelion case has the longer planetary radii from the sun,  thus a longer major axis,  and slower speeds along the ellipse.  Thus the travel time is longer.  This is almost two months different for the two bounding-limit cases.  The detailed orbital mechanics calculations are textbook stuff not given.

Note that when leaving Earth,  you want to accelerate in the direction that it orbits the sun (posigrade) to achieve the desired perihelion velocity about the sun for the transfer orbit.   Note also that you want to arrive at Mars slightly ahead of the planet in its orbit,  since your aphelion speed is less than its orbital speed.  In effect,  you want Mars to “run over you from behind”.  Any deceleration burn relative to Mars will be in the posigrade direction,  to speed you up about the sun,  in order not to be run over so fast by Mars from behind.

Departing Mars,  your escape burn will be in the retrograde direction,  to slow your velocity with respect to the sun,  down to the desired aphelion velocity.  When you reach Earth,  you will be catching up to it from behind,  since your perihelion velocity is greater than Earth’s orbital velocity.  Any deceleration burn will be in the retrograde direction,  so as not to hit the Earth from behind.

Arrival and Departure Speeds and Geometries

Figure 2 gets you from arrival and departure speeds “far” from the planet to arrival and departure speeds in close proximity to the planet,  where any propulsive burns are actually made.  These data are all planet-centered coordinates.  When still “far” from the planet,  speeds are denoted as Vinf.  When in close proximity,  speeds are denoted as Vdep or Varr.  The difference is caused by the action of the planet’s gravity on the vehicle:  if departing,  it slows you;  if arriving,  it speeds you up.  That energy-based calculation looks like this:

Vinf  =  [Vdep^2 – Vesc^2]^0.5  and Varr  =  [Vinf^2 + Vesc^2]^0.5,  where Vesc is planet escape speed

 Figure 2 – Arrival and Departure Speeds for Hohmann Transfer

There are departure and arrival geometries shown on Figures 1 and 2 that offset the perihelion and aphelion of the transfer orbit from exactly centering on the planetary center positions.  These are offsets on the order of 10^4 km compared to planetary orbit radii on the order of 10^8 km.  This is an error well under 0.01%,  so it is ignored for the purposes of this article.

However,  these offsets are important for entering orbit,  or for making direct landings.  This is because you want the planet’s rotation or its posigrade low orbital speed to assist your propulsive burns to achieve the necessary speeds.  Earth departures (and arrivals) will be on the side away from the sun.  Mars arrivals and departures will be on the sunward side. 

Figure 2 shows the Mars arrival delta-vee data dVorb to get from Varr to low orbit speed Vorb,  for the 3 cases.  These are the same magnitude as the delta-vees required to depart Mars orbit onto a trajectory home.  Also shown are the Earth departure delta-vee data dVorb from low orbit onto the trajectory to Mars.  These are the same magnitude as the arrival-home delta-vee data,  if recovering into low Earth orbit.  Escape and low orbit speeds are shown for both planets.  These are actually surface values.

Looking through these data,  there is little effect of the bounding cases on the Earth orbital departure dVorb data.  But there is noticeable difference between the cases for the Mars dVorb data.  If you intend to fly anytime,  then you must design for the worst cases.  For orbital departure from Earth,  that is dVorb = 3.85 km/s,  or an achieved Vdep = 11.75 km/s from the surface.  That last is how “C3” is calculated,  C3^0.5 being the Vinf to which Earth’s orbital speed adds,  for sun-centered trajectory analysis.  Mars orbital dV data range from 1.76 to 1.80 km/s.  To fly anytime,  you must design for an orbital entry burn of dVorb = 1.80 km/s.  C3 values are also shown for both Earth and Mars.

Departing From Earth

There are two ways to depart from Earth onto a Hohmann transfer orbit to Mars.  One is to depart from Earth orbit,  which requires a posigrade burn on the side of Earth opposite the sun (as shown in Figures 1 and 2).  Only timing of the burn and its pointing direction are critical.  See Figure 3 for the “jigger factors” (and where they apply) to use the velocity data and the rocket equation to size mass ratios.  This presumes two stages to orbit,  and a third stage to get you from orbit onto the transfer trajectory. 

The other way is a direct launch onto the trajectory from the surface of the Earth.  The launch window for this is very tight,  because the final direction is so critical.  That departure,  too,  needs to enter the trajectory in a posigrade direction on the side away from the sun,  to have both Earth’s orbital speed and its rotation speed help you achieve the necessary speed about the sun.  See again Figure 3.

Strictly speaking,  you want to apply the gravity and drag loss “jigger factors” to the achieved delta-vee demanded of the first stage.  The second stage operates with negligible drag and gravity losses.  Once in space,  for impulsive burns,  there are no gravity and drag losses.  Those factors are 1.00.

Most people do not have a specific vehicle in mind,  and don’t know its staging velocity.  You can still get into the ballpark very realistically,  applying the gravity and drag losses as a jigger factor (1.10) to the delta vee from surface to orbit,  and no losses to the delta vee from orbit to departure velocity.  From orbit,  all “jigger factors” are just 1.00.

The same concepts actually apply to a surface launch from Mars.  Adjust the gravity loss factor with a multiplier of 0.384,  and the drag loss factor by a multiplier of 0.007.  Add them to unity for your jigger factor.  Instead of a 1.10 jigger-up factor on Earth,  it’ll be closer to 1.02 on Mars.  You can do Mars departure with a single stage,  usually.  

 Figure 3 – Departing Earth From Either Orbit or the Surface

Arrival at Mars

Arrival at Mars can take any of three possible forms:  (1) propulsive burn in the posigrade direction to decelerate into orbit about Mars,  (2) a direct-entry aerobraking trajectory to a landing direct from the interplanetary trajectory,  and (3) multiple aerobraking passes to capture into an initially highly-elliptic orbit,  that gradually decreases its apoapsis due to aerobraking drag at succeeding periapses. 

Of those 3 ways to arrive,  one is still quite experimental and further made uncertain by the large variability of density profiles in the Martian atmosphere.  These vary from season to season and site to site by factors as large as 2.  For that reason,  option (3) repeated aerobraking passes is just not yet recommended for entering Mars orbit from the interplanetary trajectory. 

Entering Mars orbit with a rocket burn is quite repeatable.   At this time,  that is the recommended method for reaching orbit about Mars.  Its required dV = 1.80 km/s for mass ratio design purposes.  The deorbit burn is about 0.05 km/s,  also quite repeatable.  Only timing and pointing direction of the deorbit burn are dominantly important to landing accuracy.  From there,  it is entry,  descent,  and landing (EDL),  with a touchdown retro-propulsive burn. 

We do have considerable experience direct-landing probes on the Martian surface from the interplanetary trajectory.  Precise location of the entry interface point (and shallow path angle) is critical to landing accuracy.  No burn is required to do this,  excepting a final touchdown retro-propulsive burn.  You need excess touchdown burn capability,  in part to correct for the entry trajectory errors the variable density profile induces.

Entry,  Descent,  and Landing (EDL)

The characteristics of that landing depend strongly upon its ballistic coefficient.  This is object mass divided by the product of its frontal blockage area and its hypersonic drag coefficient (referenced to that same area).  Because of square-cube scaling,  massive objects inherently have higher ballistic coefficients than light ones.  2 x dimension is 2 x coefficient.  2 x mass is 2^(1/3) dimension.  This affects the altitude at end of hypersonic aerobraking,  heavy objects penetrating much closer to the surface before slowing.  See Figure 4.


Figure 4 – Mars Landings

The entry path angle must be quite shallow to raise altitude at end-of-hypersonics,  and also reduce peak heating and deceleration gees.  There is the risk of bouncing off the atmosphere at speeds above escape,  to be lost in space forever.  Some downlift capability early in the descent can reduce that risk. 

Later,  as speeds reduce,  gravity wants to bend the trajectory downward.  This risks impacting the surface before your aerobraking is done,  in the thin atmosphere of Mars.  Uplift capability later in the trajectory can reduce that risk.  But the path angle will inevitably steepen as the hypersonics end,  at local Mach 3 (~0.7 km/s) for blunt shapes.

From there,  assuming an average 45 degree path downwards,  you are but ~10 seconds from impact at ~ 5km altitude with high ballistic coefficient.  With a low ballistic coefficient,  you are nearer 20 km altitude and ~ 1 minute from impact.  

On the higher trajectory at low ballistic coefficient,  you barely have the time to wait a few seconds for further drag deceleration to Mach 2-2.5 (~0.6-0.5 km/s) and then deploy a supersonic ringsail chute (a few more seconds).  That will decelerate you to high subsonic (~0.2 km/s) in several more seconds,  but no slower than that,  in that thin air.  From there,  it is retro-propulsion to touchdown.

On the lower trajectory at high ballistic coefficient,  you have no time to wait for anything!  You must fire up retro-propulsion for touchdown from the Mach 3 (~0.7 km/s) point.  It will take thrusting at 3-4 standard Earth gees to zero that speed before touchdown.  There is no way around that.

The foldable and inflatable heat shield concepts are the means to have a low ballistic coefficient with a massive object.  These are entirely experimental,  not technologies ready-to-apply.  They have only flown once or twice,  and highly experimentally at that.  You simply cannot plan on using these yet. 

Because of the variability in the Martian density profiles,  and more importantly,  because of hover/maneuver needs to effect a safe touchdown,  I would never recommend a factor less than 1.4-1.5 be applied to the retro-propulsion dV requirement,  be it 0.2 or 0.7 km/s.  This jigger factor recommendation is also given in Figure 4.

On Earth,  the air is much thicker,  and the end-of-hypersonics altitudes much higher (~45 km).  There is plenty of time to use chutes on objects small enough not to overload them.  Otherwise,  pretty much the same basic considerations apply,  whether entering from Earth orbit,  or direct from the  interplanetary trajectory.  Only the heating is more severe for direct entry (~12 km/s at entry vs ~8 from orbit).   It was just under 11 km/s returning from the moon.  Could be rougher,  if a higher-energy transfer orbit is used.  Some designs call for 16-17 km/s at direct entry.

Design Requirements Summary

All of these requirements are based upon Hohmann min-energy transfer orbits as the trajectory to Mars,  or returning from it.  The worst-case one-way travel time is 283 days at Mars and Earth aphelion distances.  Average is 259 days.  Min at perihelion/perihelion geometry is 235 days. 

For departing directly from the Earth’s surface,  the launcher must actually achieve 11.75 km/s (relative to Earth-centered coordinates) in close proximity to the Earth.  Design dV’s for stage mass ratios will be higher.  These would be 1.10 Vstage plus (11.75-Vstage) as the total mass ratio design dV summed for all stages.  Failing a good value for Vstage,  estimate dV = 1.10 Vorbit plus Vdep – Vorbit,  summed for all stages.  Spread this across “probably 3 stages”. 

For departing from low Earth orbit,  the orbital mechanics dV is the rocket mass ratio design dV = 3.85 km/s,  figured as worst case for planetary orbit eccentricities.  Two stages are required to reach low Earth orbit.  The summed mass ratio design dV for that is 1.10 Vstage plus Vorbit - Vstage.  Failing a good value for Vstage,  summed design dV for the two stages is 1.10 Vorbit.  Either a third stage,  or refueling on orbit,  is required to get from Vorbit to Vdep.  That orbit departure dV is unfactored:  Vdep - Vorbit - 3.85 km/s. 

For propulsive deceleration into low Mars orbit,  the worst case dV = 1.80 km/s,  orbital mechanics dV being equal to actual rocket mass ratio design dV.    If a higher-energy trajectory,  this value is higher. 

For Mars orbital-based missions,  the de-orbit dV = 0.05 km/s. 

For both orbit-based and direct-entry landings on Mars,  the high-ballistic coefficient touchdown dV = 1.05 km/s.  The low-ballistic coefficient touchdown dV with chutes is 0.3 km/s.  These are rocket mass ratio-design values,  already factored by 1.50. 

For departing directly from the Mars’s surface,  the launcher must actually achieve worst-case Vdep = 5.35 km/s (relative to Mars-centered coordinates) in close proximity to Mars.  Design dV’s for stage mass ratios will be higher.  Estimate summed dV = 1.02 Vorbit plus Vdep – Vorbit,  summed for all stages.  Spread this across “probably 2 stages”. 

For departing from low Mars orbit,  the orbital mechanics dV is the rocket design dV = 1.80 km/s,  figured as worst case for planetary orbit eccentricities.  Only one stage is required to reach low Mars orbit at mass ratio design dV = 1.02 Vorbit = 1.02*3.55 km/s. 

For direct entry at Earth from the interplanetary trajectory,  no propulsive burn is required,  but entry occurs above escape speed at 11.75 km/s or higher (depending upon the trajectory).  Depending upon the size of the vehicle,  and whether it can land with “only chutes”,  a final retro-propulsive touchdown burn under dV = 0.1 km/s may be required.   Or it could be as high as 1 km/s,  if chutes are infeasible. 

For recovery into Earth orbit from the interplanetary trajectory,  a dV = 3.85 km/s burn is required.  This orbital mechanics requirement is equal to the mass ratio-sizing requirement.  From there,  if landing is required,  the deorbit burn is on the order of 0.1 km/s.  There may,  or may not,  be a touchdown burn requirement,  that could vary from 0.1 to 1.0 km/s,  depending upon whether chutes are feasible. 

It might be possible to use multiple aerobraking passes to capture into an elliptical orbit at Earth,  and let the repeat passes reduce apogee altitude by drag instead of rocket burn.  This requires either a small apogee burn to stabilize the orbit outside the atmosphere,  or else a burn to control where the landing occurs.  Either might be on the order of 0.1 km/s.   

If landing,  there might be a touchdown burn between dV = 0.1 and 1.0 km/s,  depending upon whether chutes are feasible.  This aerobraking capture still needs demonstration and development,  but does not suffer from the factor 2 variability of Martian atmosphere density.  It might be proven feasible at Earth sooner than at Mars.