Tuesday, April 17, 2018

Reverse-Engineering the 2017 Version of the Spacex BFR

Update 9-24-18:  Spacex has revealed a new finned version of the BFS second stage.  Much useful data is still missing,  but I did reverse-engineer much of its performance.  That is posted here on this site,  as an article dated 9-24-18,  and titled "Relevant Data for 2018 BFS Second Stage"


I visited the Spacex website in April 2018 and recovered from it the revised characteristics of the large BFR Mars vehicle,  as presented during 2017.  These characteristics are somewhat downsized from those in the 2016 presentation given in Guadalajara.  Thus this article supersedes the previous article posted on this site 10-23-17,  titled “Reverse Engineering the ITS/Second Stage of the Spacex BFR/ITS System”.  Both articles share the search keywords “launch”,  “Mars”,  and “space program”.  The older article has been updated to redirect readers here.

Data From the Spacex Website

What is listed includes a basic vehicle diameter of 9 meters (both first stage booster and second stage spacecraft),  a booster length of 58 m,  and a spacecraft length of 48 m.  The forward half of the crewed spacecraft includes an unpressurized cargo bay,  and over 825 cubic meters of pressurized space for the crew and passengers.  The rest is propellant tankage and engines. 

No data are given for the uncrewed cargo version,  but it appears quite similar overall,  with a giant clamshell door to open the entire space corresponding to the crewed vehicle’s unpressurized and pressurized payload spaces.

For the crewed vehicle,  payload mass is listed as 150 metric tons to low Earth orbit,  and the same on to Mars with refilling on-orbit.  Return payload (presumably from Mars) is listed as 50 tons,  with refilling done on Mars by in-situ propellant manufacture from local ice deposits and atmospheric carbon dioxide.  The cargo deliverable to orbit by the uncrewed cargo version is also said to be 150 tons.

The crewed spacecraft is listed as having an 85 ton inert structure weight.  Musk says the data right now actually say 75 tons,  but that always grows as development proceeds.  Propellant loadout in the tanks is 240 tons of liquid methane,  and 860 tons of liquid oxygen (1100 total),  both superchilled to a higher density than in normal usage.  The listed data imply similar numbers for the uncrewed cargo version.

Basic weight statements for the crewed vehicle (implied to operable either manned or unmanned by the mission planning) would then be as follows (all masses are metric tons):

Item                     at launch             ign. in LEO           ign. on Mars
Payload                150                         150                         50
Inert                      85                           85                           85
Dry                      235                         235                         135
Prop.                  1100                       1100                       1100
Ign.                    1335                       1335                       1235

 There are not enough data given to establish a weight statement for the first stage booster,  because no inert weight was given.  Only the gross takeoff mass (for 150 tons of payload) was given to be 4400 metric tons.  The inert fraction of the crewed spaceship is a bit over 6%.  Falcon-9 first stages seem to be just under 5%. 

On the assumption that the inert mass fraction of the BFR first stage is 5%,  given the 1335 ton max weight of the second stage,  then the first stage inert mass should be near 220 metric tons.  Under that assumption,  its weight statement would be approximately:

Payload (second stage at max)                   1335
Inert                                                              220
Dry                                                               1555
Propellant                                                    2845
Ignition                                                        4400

The first stage is listed as having some 31 sea level Raptor engines.  The second stage spacecraft is listed as having 2 sea level Raptor engines of high gimballing capacity,  and 4 vacuum Raptor engines of limited gimballing capacity.  The two versions differ only in the expansion bell size,  very much like the Merlin engines used on the Falcon 9 and Heavy. 

Design chamber pressure is listed as 250 bar,  with growth eventually anticipated to 300 bar.  The engines are said to be throttleable from 20% to 100% of rated thrust.  The given specific impulses (Isp) apply to 100% thrust at 250 bar,  and should decrease slightly as thrust is throttled down.  No data are given for that Isp reduction.  It is a relatively minor effect at this level of analysis,  and is ignored in my reverse-engineering analysis here.

Data as listed for the individual Raptor engines are as follows.  The sea level design data as presented do not report vacuum thrust,  but an exit diameter is given,  so that it may be calculated using sea level air pressure of 101.325 KPa.  I did this in my reverse-engineering analysis.  Only vacuum performance is listed for the vacuum design.  Listed data are for one engine. 

The data on the site claim a vacuum engine exit diameter too big to fit a 9 m diameter vehicle;  that has to be a typographical error!  Because the vacuum design is never operated with backpressure (excepting 6 mbar on Mars,  essentially negligible),  the lack of a reliable vacuum exit diameter does not impact my estimates.  

Version                                sea level                               vacuum
Vacuum Isp, s                     356                                         375
Vacuum thrust, KN          not given                                 1900
Sea level Isp, s                   330                                         not given
Sea level thrust, KN         1700                                         not given
Exit dia, m                           1.3                                        12.4 (typo !!!)

The presentation as given on the Spacex website does show the reentry sequence as currently envisioned at Mars.  This shows direct entry from an interplanetary trajectory,  with an entry interface speed up to about 7.5 km/s.  Initially the vehicle is flown inverted for downlift,  presumably to keep it from “bouncing off” the Martian atmosphere while traveling faster than Martian escape speed.  Later in the entry trajectory,  the vehicle rolls upright for uplift, in order to shape the trajectory. 

By the time the hypersonics are ending at about 0.7 km/s (local Mach 3) speeds,  the vehicle is within 4-5 km of the surface,  and climbing upward towards about 10 km altitude.  It reverses to tail first for the retropropulsive burn,  all the way to landing.  The presentation claims max 5 gees deceleration during entry sequence while hypersonic.

No data is given at all for the return entry at Earth.  Presumably,  this would also be direct from the interplanetary trajectory,  with entry interface speeds somewhere in the vicinity of 17 km/s.  This will be a very high-gee entry,  as return from the moon with Apollo was 11 gees at 11 km/s entry interface speed.  12-15 gees (or more) is nothing but a gut-feel guess.

The data on the site show 4 landing legs whose span is 3 or 4 times less than the length of the crewed second stage vehicle.  Cargo is lowered to the ground from the cargo bay door with a crane.  People would leave the vehicle by the same means,  and reverse the crane to return to it.

Organizing the Analysis

The Spacex website presentation claims refilling on orbit from uncrewed tanker vehicles.  The illustrations show variously 4 to 5 such tankers,  but no data are given.  Propellant transfer is by tail-to-tail docking,  with thruster-induced microgravity driving propellant flow from one vehicle to the other.  It is unclear whether the tanker is an unmanned crewed vehicle or the cargo vehicle,  and it is unclear whether propellant is loaded as cargo,  or is leftover in the tanks after flying to orbit with no payload.

The tanker performance problem is not analyzed here.  

The analyses here were made using simple rocket equation estimates modified with realistic “jigger factors” for gravity and drag losses.  Because of the long times in interplanetary flight with cryogenic propellants,  other “jigger factors” get included to account for boiloff effects and midcourse correction budgets.  And because retropropulsive landings must have a margin to adjust touchdown if obstacles are encountered,  yet another “jigger factor” must be included to model the need to hover and/or redirect touchdown laterally some distance away. 

For Earth launch with slender,  “clean” vehicle shapes,  something like 2.5% gravity loss and 2.5% drag loss provide realistic first estimates.  The two add to 5%,  which gets incorporated into a “jigger factor” of 1.05,  that multiplies the kinematic delta-vee requirement,  making it suitably larger to cover gravity and drag losses in the simple rocket equation.  At Mars,  gravity is weaker and the air much thinner,  so I reduce the gravity percentage by a factor of 0.384 (Mars surface gravity in gees),  and the drag percentage by a factor of 0.007 (Mars surface density ratioed to Earth standard).  In retropropulsion,  drag helps rather than hinders,  so the two percentages subtract instead of adding. 

I assumed a factor of 1.1 increase on the already “jigger-factored” propellant weight (out of the rocket equation for that burn) to include a “kitty” for minor midcourse corrections,  and the same for long-term propellant boiloff effects in transit.  I also assumed a hover/redirect factor of 1.5,  to multiply the same already-factored propellant weight,  during landing burns. 

These calculations can be made by hand,  but are very conveniently entered into a spreadsheet,  for rapid changes and refinements.  That is what I did for this analysis.  Its results were incorporated into a series of figures.

First Stage Problem

The first stage problem has but one payload,  a weight statement based on the 5% inert fraction assumption,  and suitable sea level and vacuum engine performance data for the sea level Raptor engine design.  The website presentation does not give any indication of the staging speed or altitude,  only depictions showing the trajectory bent over almost level by the time staging occurs. 

This is complicated by the need to fly the booster back to launch site,  and land it there,  very much like the Falcon-9 first stages.  Therefore,  beyond just reaching staging speed against realistic gravity and drag losses,  enough propellant must remain on board after staging,  for the stage (with no payload) to more-than-“kill” its flight velocity with a boostback burn,  ease the entry heating with a shock-penetrating entry burn,  and conduct the final touchdown burn. 

I made the “reasonable” guesses of allowing 0.99 km/s worth of delta-vee for the touchdown burn,  an arbitrary 0.10 km/s delta vee for the entry burn,  and a boostback delta-vee equal to,  or slightly exceeding,  the speed at staging.  I assumed staging altitude to be essentially in vacuum,  so that the average of the sea level and vacuum Isp’s could be used to represent average booster performance. 

Staging speed was an input assumed value,  since it was not specified in the presentations on the website.  I iteratively modified this value until the 3 flyback burns all had reasonable delta-vee values.  The boostback kinematic delta-vee is the staging speed.  It is figured for the ascent weight statement,  “jiggered” by factor 1.05,  and then plugged into the rocket equation for a mass ratio less than what the stage provides overall.  Dividing launch mass by that ratio gets the burnout mass,  and the difference is the ascent boost propellant mass burned to reach staging speed.

Deleting the payload mass (second stage) gives a new weight statement for booster inert and the mass of propellant still on board.  It corresponds to a certain delta-vee from the rocket equation,  “jigger-factored” down by 1.025,  since there is no drag outside the atmosphere.  Subtracting 0.99 km/s for the touchdown burn,  and 0.10 km/s for the entry burn,  leaves the delta-vee benefit available for the boostback burn.  I adjusted my assumed staging speed until my boostback delta-vee equaled or slightly exceeded this staging speed value. 

At launch,  the sea level thrust of the 31 sea level Raptor engines totals to 5375 metric tons-force (force in KN/9.805).  The website data lists 5400 tons of thrust,  close enough.  The thrust/weight ratio minus unity gives the kinematic acceleration straight upwards,  in gees.  At staging,  one must correct to vacuum thrust of the sea level engines,  and since the trajectory is nearly level,  the thrust/weight ratio is the pathwise kinematic acceleration in gees.  These values can be factored by the throttle percentage expressed as a fraction,  if needed.  I bounded the burnout accelerations by calculating values at 100% and 20% thrust.

Those results are summarized in Figure 1.

 Figure 1 – Results for the Booster Problem with Flyback Recovery

Going to Mars After Refilling in Earth Orbit

This is not quite straightforward,  because you cannot use all your propellant to go to Mars.  You must have enough propellant still on board after departure,  to enable the landing on Mars.  That means you analyze the Mars landing first,  and then the departure burn from Earth orbit. 

You have to analyze from dry tanks at touchdown on Mars back to the landing burn ignition conditions,  complete with Mars retropropulsion gravity/drag effects,  for the min propellant needed to land.  Then you scale that amount of propellant up with the hover/redirect factor on your delta-vee for a realistic propellant budget to land from the rocket equation.

Then you scale that realistic landing budget up with the boiloff factor and the midcourse factor to find the actual reserve propellant that you must still have on board after the departure burn.  That is what sets your weight statement for the departure burn. 

These results for the Mars landing are given in Figure 2.

 Figure 2 – Results of the Analysis of the Mars Landing

The departure burn from Earth orbit sees an Earth gravity loss,  but no drag loss.  The usable delta-vee from it adds to the Earth orbit velocity for a velocity Vdep at that Earth orbit location.  A little farther from Earth,  you figure a velocity-at-infinity as Vinf = (Vdep^2 + Vesc^2)^0.5.  Then you add Earth’s orbital speed to that value for the vehicle speed with respect to the sun Vwrt sun. 

For a min energy Hohmann-type transfer ellipse orbit there is a perigee velocity,  which varies with planetary positioning along their ellipses.  I used the worst case (highest) value.  The vehicle velocity with respect to the sun Vwrt sun,  minus the Hohmann transfer perigee speed Vper-HOH,  is the margin you have,  that might be used to fly a faster trajectory.  Hohmann transfer is about 8.5 months one way. 

These results for the departure from Earth orbit are given in Figure 3.

Figure 3 – Results for the Analysis of Departure from Earth Orbit

Returning From Mars

Very similarly to the trip outbound to Mars,  one must analyze first the landing on Earth,  to define the propellant needed to land there,  jigger that up for the transit,  and have that quantity in reserve after the Mars departure burn.  So you analyze the Earth landing first,  then the departure from Mars,  which is a direct ascent into the interplanetary trajectory.

I used the same basic end-of-hypersonics at 0.7 km/s (local Mach 3) as my kinematic definition of min landing delta-vee.  This just happens at higher altitude,  and in much thicker air,  on Earth.  Unlike Mars,  it is quite feasible to fall a long way in the transonic/low-supersonic flight speed range,  before reversing vehicle orientation to tail-first for the touchdown burn. 

You “jigger-up” this min delta vee by a gravity-drag factor (near 1 in retropropulsion on Earth) and by the hover/redirect factor (I used the same factor 1.5 for this).  That much larger delta-vee goes through the rocket equation for a mass ratio from dry tanks at touchdown.  This leads to a realistic landing propellant budget.  Then you factor that up for boiloff and midcourse,  to find the reserve propellant that must be on board after the Mars departure burn. 

The Earth landing results are given in Figure 4.

 Figure 4 – Results for the Earth Landing Analysis

This Earth landing propellant budget at Mars departure then adjusts the weight statement of the refilled craft upon Mars,  in addition to the stated reduction in return payload.    This is a direct departure:  the weight statement and choice of engines finds the ideal delta-vee,  adjusted downward by the Mars gravity-drag factor to a realistic delta-vee.  We are not stopping in Mars orbit,  this realistic delta-vee becomes the speed of the vehicle near Mars.  It is adjusted using Mars escape velocity to find the Vinf in the vicinity of Mars.  That is in turn subtracted from Mars’s orbital velocity to find the velocity with respect to the sun.  This is compared to the Hohmann min energy transfer orbit’s apogee velocity,  to determine any margin available for a faster trip home. 

These numbers indicate little or no potential for a fast return home.  The results are given in Figure 5. 

Figure 5 – Results of the Mars Departure Analysis


Using Spacex’s own data  plus some reasonable assumptions regarding gravity and drag losses,  and hover requirements for retropropulsive landings,  and for boiloff and midcourse budgets,  I calculated performances estimates,  for the big Spacex Mars vehicle as presented in 2017, that are not very far at all from what is claimed in their 2017 presentation. 

I show some potential for a slightly-faster trajectory to Mars than min energy Hohmann transfer.  I show very little,  essentially zero,  potential for a faster return trajectory from Mars,  compared to min energy Hohmann transfer. 

To get these data,  I had to assume that the inert mass fraction of the BFR first stage is 5%,  and I had to assume that the BFR stages at just about 2.55 km/s flight speed,  outside the sensible atmosphere,  and already almost level. 

I used the reported engine performance data for the sea level and vacuum forms of the Raptor engine,  operating at 250 bar chamber pressure at full thrust.  If the chamber pressure can be raised closer to 300 bar (as Spacex wants),  some of the faster-trip performance shortfalls ease.  I did not analyze these data to quantify that effect.

Gravity and drag loss effects upon ideal rocket equation delta vee are assumed at 2.5% each here on Earth,  and ratioed down by 0.384 for gravity,  and 0.007 for drag,  at Mars.  Propellant quantities coming from the rocket equation mass ratio and appropriate weight statements get ratioed by an assumed factor of 1.5 for retropropulsion hover/redirect effects,  by a factor of 1.1 for boiloff effects in transit,  and by a factor of 1.1 for budgeting midcourse correction propellant. 

To correct sea level thrust of a sea level Raptor engine to vacuum conditions,  add a force equal to the exit area multiplied by Earth sea level air pressure.  This does not affect rocket equation results,  but it does affect vehicle acceleration-capability calculations.  These help you choose which engines to burn,  and what levels of throttling to use.

Spacex has posted data for anticipated Mars entry from the interplanetary trajectory,  but not for Earth entry from the interplanetary trajectory.  It is peak entry deceleration gees during the return to Earth that is very probably the highest gee requirement for occupants to endure.  This is not something controllable with engine thrust,  as there is no propellant available to budget for this purpose. 

Earth entry deceleration from Mars (on a direct entry from the interplanetary trajectory) is quite likely to be far more severe than the Apollo 11-gee peak deceleration coming back from the moon at an entry interface speed of 11 km/s.  Coming back direct from Mars,  entry interface speed is likely to be in the vicinity of 17 km/s.  The crew simply must be very physically-fit to endure this.  This is a serious issue yet to be addressed in the Spacex presentations.

Landing stability of a relatively tall and narrow vehicle,  on unprepared rough ground on Mars,  is not addressed here.  This is another serious issue yet to be addressed in the Spacex presentations.

Update 4-18-18 artificial gravity

For Spacex’s mission plan with its BFR vehicle,  the health risk for high gee entry occurs at Earth return.  It cannot be avoided.  The occupants so exposed will have endured months-to-years of low Mars gravity (0.384 gee),  followed by about 8 to 8.5 months exposure to zero-gee on the transit home.  They are very unlikely to be physically fit for an 11+ gee entry,  even if Mars gravity is found to be fully therapeutic for microgravity diseases.   

The support for that assertion comes from years of orbital experiences at zero gee.  Astronauts exposed to zero gee for times on the order of 6 months to a year have proven to be fit enough to endure a 4 gee ride down from Earth orbit,  with an entry speed of 8 km/s.  We have absolutely nothing to point at,  to support the assumption that higher gee levels are safely endurable in that physical state!   Coming from Mars is a faster entry at about 17 km/s than from the moon,  and that was an 11 gee ride at 11 km/s.

Given that risk,  artificial gravity for at least the voyage home seems prudent.  This could possibly be accomplished by having two ships make the return voyage together.  Taking advantage of the refilling plans and procedures,  dock the two ships tail-to-tail during the long coasting transit.  Spin them up end-over-end with the attitude thrusters.  At the nominal 4 rpm spin limit,  near-Earth gee levels are obtained as shown in Figure 5. 

Figure 5 – Obtaining Spin Gravity in Two BFR Ships Docked Tail-to-Tail

The 4 rpm limit is a “fuzzy” limit.  If a very slightly-higher spin rate (maybe 4.6 or 4.7 rpm) is tolerable to the balance organs in the middle ear,   then very near full Earth gravity can be simulated in the occupied decks.  This is also shown in Figure 5 as the data in parentheses.  This is an artifact of the size of the BFR vehicle.  Achieved gee level is proportional to spin radius,  and to spin rate squared.  A nominal reference point is 1 gee at 56 m radius and 4 rpm.

There are two inconvenient design issues with this notion,  but they are not “show-stoppers”.  One is the reversed directions for up and down,  sitting on the landing legs versus spinning for artificial gravity.  What were floors become ceilings,  and vice versa.  All the interior equipment and appurtenances will have to be reversible physically,  or “double-ended” if not. 

The other is the solar panel fans that Spacex shows for powering these vehicles with electricity.  The presentations show them deployed near the tail of the vehicle while in space (free fall).  These will have to be strong enough to deploy properly,  and stay in position,  while exposed to low levels of effective gravity while spinning.  As shown in Figure 5,  these levels of gravity will be less than lunar gravity.  Capture of solar energy for conversion to electricity will usually be intermittent,  at the spin rate.

These are design inconveniences,  to be sure.  But in comparison to losing occupants due to heart failure at high entry gee,  just minutes from returning to Earth,  these are minor inconveniences.  I am fond of reminding people that “there is nothing as expensive as a dead crew”. 

Update 4-19-18 landing stability etc:

Landing stability on Mars is associated with overturning issues,  landing pad penetration into soil,  and the perturbing dynamics of trying to land among rocks.  There are also issues with the jet blast flinging debris where it is not wanted,  although these only arise with subsequent ships landing near the first ship or any other structures or equipment already there. 

From an analysis standpoint,  there are static effects and dynamic effects.  From a mission standpoint,  there is a landing at low weight,  and a takeoff at high weight.  For the takeoff,  there are few,  if any dynamic effects to worry about.

Landing Statics

This topic divides into static overturn stability,  and the soil bearing pressure underneath the landing leg pads.  Static overturn stability simply requires that the weight vector fall within the polygon defined by the landing pads,  no matter how off-angle the ship sits,  such as on inclined ground.  As shown in Figure 6 below,  this isn’t much of an issue for inclinations oriented directly toward a pad,  as illustrated. 

The center of gravity position in the figure is only a guess,  but a realistic one.  The span pad-to-pad is only a guess,  but also a realistic one.  The ground could incline some 18 degrees directly toward a pad,  and still be stable,  as shown in the figure.   If the inclination is directly between two pads,  the lateral distance is 70% of the value shown,  for a max inclination angle of about 13 degrees.

13 degrees is a rather steep local slope on terrain chosen to be flat.  We can conclude that the otherwise tall BFR spaceship is at relatively lower risk of simple static overturn,  as long as small localized hazards like a pad coming down in a dry stream gully can be avoided.  That might be very challenging to satisfy in a robotic landing,  though.  Hopefully the available stroke in each landing leg exceeds the roughly 2 m shown in the figure.  That should take care of most of the localized roughness hazards.  Stroke rate capability should be comparable to ship speed just as it touches down.

At landing,  with the full 150 ton payload and 85 tons of inert,  the ship at “dry tanks” masses 235 metric tons.  If the landing is “perfect” and does not use any of the hover/redirect allowance built into the propellant budget,  there might still be something like 53 tons of propellant on board at touchdown,  bringing the as-landed mass to 288 tons as a maximum.  At Mars 0.384 gee,  the corresponding weight on the landing legs is 1084 KN. 

If evenly distributed among the 4 landing legs,  and if the total pad area is 10 sq.m as shown in the figure,  then the bearing pressure exerted upon the soil after landing is 108-109 KPa.  If all the propellant is used,  the bearing pressure is the lesser 88.5 KPa shown in the figure. 

The figure shows typical soil bearing pressure capabilities for two soils that might be like soils that could be encountered on the plains of Mars.  One is soft fine sand,  like many deserts with sand dunes on Earth,  capable of from 100-200 Kpa.  The other is more like desert hardpan on Earth,  with lots of gravel mixed into coarse sand and relatively-compacted:  some 380-480 KPa.  Excluding dynamic effects,  even the soft fine sand seems capable of supporting the low weight of the as-landed ship.

Landing Dynamics

If there is some inclination,  it will tend to throw some of the ship’s weight toward the downslope pad or pads.  Landing impact dynamics could possibly double the static forces on a short transient.  If we double the static bearing pressures,  this should typically “cover” the landing impact dynamics and any small inclination effects.  For a ship with residual propellants on board,  the doubled static bearing pressures are in the 220 KPa class. 

That rules out soft fine sand by considerable margin.  Any landing site must be desert hardpan or better in terms of soil bearing strength,  or else the landing pads had better total far more than 10 square meters of bearing area (something difficult to achieve).  Thus it would pay to select a landing site already visited by an earlier probe or rover,  whose visit could verify soil type and estimated strength. 

Once the equipment is in place to prepare hard-paved landing sites ahead of time,  this restriction loosens.

About the worst conceivable landing dynamics event is for one pad to touch down on a boulder,  and then slip off during the touchdown,  leaving the vehicle temporarily unsupported on that side.  If that happens to be downslope,  the vehicle will start to topple that way,  while the leg strokes to reach the actual surface. 

If one assumes a realistic 5 degree local slope down toward the pad that hits and slips off a 1 m boulder,  then for the max landed weight of 1084 KN (as calculated above),  a side force at the center of gravity of about 95 KN acts to topple the vehicle.  This is on an effective moment arm of 22 m,  using the surface as the coordinate reference.  That torque is 2090 KN-m = 2.09 million N-m.

Approximating the vehicle as a solid bar 48 m long of mass 288 metric tons,  its moment of inertia is roughly (1/12) m L^2 = 55.3 million kg-sq.m.  The resulting angular acceleration is something like 0.0378 rad/sq.sec or 2.16 deg/sq.sec. 

Ignoring the recovering stroke rate of the slipped leg,  the vehicle could rotate through about 5 degrees for that pad to actually strike the real surface.  Adding the inclination of 5 degrees,  that’s 10 degrees out-of-plumb,  within the 18 degree limit for one pad directly downslope,  and also within the limit for two pads downslope. 

Now,  in this transient as the slipped pad hit surface,  the vehicle is already moving,  and is going to take time to stop moving once the pad is on solid ground to resist.  Crudely speaking,  the time for the pad to strike the surface is near 2.15 sec,  and the ship will be moving at about 4.6 deg/s.  The pad support force has to stop this motion in its 1 remaining m of stroke. 

That pad force will be in the neighborhood of 279 KN,  just ratioed from the disturbing force by the ratio of moment arms,  and acting upon about 2.5 sq.m for one pad.  The vehicle will move another 5 degrees during this deceleration transient.

At the end of this transient,  the vehicle is about 15 degrees out-of-plumb,  dangerously close to the 18 degree limit directly toward 1 pad,  and beyond the 13 degree limit directly between two pads.  The bearing pressure under the pad exerting the restoring force is in the neighborhood of 112 KPa plus the allocated static weight pressure for the vehicle at rest (109 KPa):  a total near 221 KPa. 

From an inclination standpoint during this transient,  the vehicle is dangerously close to toppling over,  even if the soil is infinitely strong and hard.  If the soil resembles packed coarse sand with gravel,  it is theoretically strong enough to resist serious compaction under the pad exerting the restoring force,  but whatever compaction does occur allows the vehicle to incline just that much more.  If the soil resembles more soft fine sand in strength,  the restoring pad will inevitably dig in,  without being able to exert enough restoring force,  and so the vehicle will indeed topple over and be destroyed. 

The toppling risk is high if a pad strikes an obstruction of any significant size (in this example,  a boulder 1 m in dimension).  Things of this size are difficult to observe by remote sensing from orbit.  The risk situation is quite similar to the boulder field encountered during the Apollo 11 moon landing,  and requires a hover and redirect action away from the threat.

The BFR vehicle obviously needs some sort of “see-and-avoid” capability at touchdown.  This would be direct vision with a human pilot,  or something very sophisticated indeed built-in for a robotic landing.  If the vehicle were not so tall,  the moment arm of the disturbing force would not amplify the toppling effect so much.  That is why the Apollo LM and all the Mars landers have had a height/span ratio under 1.   That number is 3+ for this vehicle.

Figure 6 – Data to Support Statics and Dynamic Calculations

Takeoff Statics (Only)

At takeoff,  payload is reduced to 50 tons,  inert is the same 85 tons,  and the refilled propellant load is the maximum 1100 metric tons.  The vehicle mass at takeoff is then 1235 metric tons.  At Mars 0.384 gee,  the vehicle weight is near 4650 KN.  Evenly distributed onto a nominal total of 10 sq.m of pad area,  the exerted bearing pressure is 465 KPa.

The soil upon which it rests had better not be similar to fine soft sand,  or the landing legs will sink deeply into the soil as propellant is loaded. 

This could lead to landing leg damage as the ship launches,  or even prevent its launch,  given the high extraction forces trying to pull out deeply-embedded pads.  If the soil properties vary on a 5 m scale,  the legs could embed unevenly.  That could risk toppling over with propellants on board,  leading to a huge explosion.

If the soil resembles gravel embedded in packed coarse sand,  the static bearing pressure at launch falls within the range of soil bearing strengths,  before the safety factor of 2 is applied.  That means it is very likely that the pads will sink a bit into the soil,  although not likely enough to run the embedding risk.  If the soil properties vary on a 5 m scale,  this sink-in will be uneven,  adding to the vehicle inclination. 

Given those outcomes,  it would be wise to design larger pads on the landing legs,  or use more than the 4 legs depicted in the website presentations.   This size and shape of vehicle is more safely operated from a very level,  hard-paved pad,  or from level,  flat solid rock. 

I do recommend using the coarse sand/gravel soil strength as representative of the sand/rock mix we see on Mars.  The minimum Earthly strength of that material is about 380 KPa,  with around 480 KPa as the max strength.  I also recommend using at least a factor of 2 upon exerted pad pressures to model inclination and sudden-impact effects on landing.  Best practice would apply that to takeoff as well,  although perhaps a factor nearer 1.1 might be adequate. 

The landing leg and pad depictions on the website are very generic and lack detail.  I don’t believe this part of the vehicle has yet received much in the way of design attention.  I presumed 10 sq.m of pad area,  and it is really not enough.  This part of the design needs such attention,  as it will be difficult to incorporate 14+ sq.m of flat pad area on the ends of 4 landing legs,  and still stow these away successfully for hypersonic entry aerobrake events.  If circular,  that 14 sq.m of pad area corresponds to pads about 2.1 m in diameter.  Even my 10 sq.m analysis implies circular pads 1.8 m diameter.

Debris Flung By Jet Blast

The force exerted on the surface by the rocket streams as the vehicle touches down is just about equal to the engine thrust.  At landing on Mars,  this would be the vacuum thrust of two sea level Raptor engines.  This is about 3670 KN.  It would be exerted over an area on the ground comparable to the exit area of the engines,  or maybe a little more.  That would be something in the vicinity of 3 sq.m.  That effective average pressure is near 1220 KPa. 

That pressure is so far above the soil bearing strength,  even for coarse sand with gravel,  that the sand will get flung as a supersonic sandstorm,  and rocks of substantial size are going to get torn loose and flung with considerable force,  albeit at a rather low launch angle. 

For a 10 cm rock,  the cross section area is about 0.00785 sq.m.  The jet blast pressure acting on that area produces a force in the vicinity of 9.6 KN.  At specific gravity 2.5,  such a rock should mass something in the vicinity of 1.3 kg,  and on Mars would weigh something like 4.9N.  The force to weight ratio is huge at nearly 2000 to 1. 

Using impulse and momentum,  with a wild-guessed action time on the order of 0.01 sec,  the thrown velocity of this rock would be on the order of 75 m/s.  If the elevation angle were 10 degrees,  the range at which the rock comes down would be around 0.5 km.  These crude estimates could easily be too conservative. 

Loss of a Ship Leading to Explosion

If a ship should topple over and explode,  or crash and explode,  large debris will be flung with incredible force.  Based partly on half the speed of the military fragment impact tests,  the velocity of such debris might be in the vicinity of 1.2 km/s.  It will leave the scene at all elevation angles from zero to straight up.  In the low gravity on Mars, such debris flung at 45 degrees will travel over 350 km.

It will be almost impossible to protect from debris flung nearly vertically,  which will come down close to the site.  It would be possible to intercept the low-angle debris with an earthen embankment bulldozed around the landing pad.  That might protect neighboring ships on adjacent pads,  or base buildings erected nearby,  acting as a shadow shield.  Based on what I have seen published about the base to eventually be constructed with these BFR ships,  I don’t think this issue has yet been considered at all.  Yet,  such an explosion is inevitable over the long haul,  even if very rare. 

Conclusions and Recommendations

It is probably too late in the design process to adjust vehicle length and diameter to a shorter,  fatter proportion.  That means the landing pads must exceed about 14 sq.m total area in actual contact with the ground,  if my soil strength estimates have any reality at all.

I strongly recommend to Spacex that they start looking closely at these issues of landing and takeoff statics,  and the landing dynamics.  From what has been published so far,  I have to conclude that they have not addressed these particular issues in any detail,  yet. 

I would definitely,  and very strongly,  recommend that BFR landing sites be at least 1 km away (preferably 2+ km) from other grounded ships or any other structures or equipment,  to mitigate the flung rock hazard.  This would be true until such time as hard paved landing pads can be constructed.  “Hard” means materials with compressive and shear strengths both exceeding about 1500 KPa = 1.5 MPa.  Such will automatically satisfy needed bearing strength.

I would also recommend to Spacex that they begin considering the possible effects of a ship explosion upon adjacent ships,  base buildings,  and other nearby equipment.  Berms around hard-paved pads are highly recommended.  Such high-velocity debris will travel a long way in Mars’s low gravity,  and some of these pieces will be quite large.   Such events may actually be exceedingly rare,  but the results are unacceptably catastrophic,  no matter how rare.  Probability x cost is NOT the way to judge this.

Update 4-21-18 speculations on the tanker issue:

This update presents very speculative numbers I ran trying to understand the tanker vehicle issue.  This includes both the design of the tanker,  and how many are needed to fully refill a crewed BFR second stage vehicle in Earth orbit. 

Spacex presents on its website estimated weights data for the crewed vehicle,  and not for the cargo-only unmanned vehicle (or the tanker).  The implication is that the cargo vehicle overall characteristics are similar to the crewed vehicle.  There is no clue given as to the identity or characteristics of the tanker. 

It would make sense that the same engine section and propellant tank section would be used in both (or all three),  just with different forward section structures,  although with the same heat shield.  That is the “justification” for using the same inert weights,  propellant weights,  and payload weights for crewed and cargo vehicles in the original portion of this article.  In particular,  the inert weights are likely not to be the same,  although they are likely to be crudely similar.

Here in this update,  I extend that inert weight assumption to any potential third configuration that would serve as a dedicated tanker.  It would have additional tanks holding 150 tons of propellant mounted in the forward section,  and plumbed into the other tanks.  The other competing idea is to fly a crewed BFR unmanned,  or a cargo BFR,  but both with zero payload on board.  If fully loaded with propellant in the tanks,  either of these would arrive on orbit with considerable unused propellant beyond the landing budget (effectively the “tanker load”).

All the assumptions and engine performance data are the same as I already used in the original portion of this article.  The same basic mission analysis consideration applies:  hold enough propellant in reserve to land.  The difference here is that the landing is made with zero payload on board,  which reduces the necessary budget for landing propellant.  Plus,  no budgets need be maintained for long-term boiloff effects, or for deep space midcourse maneuvers.  Thus the reduced figure of 32.6 tons of landing propellant will suffice to land at zero payload,  even with factor 1.5 on the landing delta-vee to cover any “hover and redirect” effects. 

Similar to what I did in the original portion of the article,  we start with the landing to determine that landing propellant budget.  With all three designs sharing the same overall weight statement,  all three then share the same dry-tanks weight at landing,  and thus the same landing propellant budget.   Then we apply that reserve to each of the three tanker configuration candidates to determine how much deliverable propellant they can provide on orbit.  Assumptions are shown in Figure 7,  results in Figure 8.

 Figure 7 – Assumptions Made to Model the Tanker Problem Two Ways

Figure 8 – Results Obtained Modeling the Tanker Problem Two Ways

Applicable overall weight statements (metric tons) are:

                                crewed                 cargo                     ded. tanker
inert                      85                           85                           85 (#1)
payload                0 (#3)                    0 (#3)                    150 (#2)
b.o.                        85                           85                           235
propellant           1100                       1100                       1100
ign                          1185                       1185                       1335

#1. incl extra tanks
#2. payload is propellant
#3. capable of 150 tons, flown here at 0

The resulting as-flown weight statements are:

                                Crewed                cargo                     ded.tanker
Ign                          1185                       1185                       1335
Asc.prop              925.6                     925.6                     1042.8
b.o.                        259.4                     259.4                     292.2
less del.prop      141.8                     141.8                     174.6 (#1)
land.ign.               117.6                     117.6                     117.6
land.prop            32.6                        32.6                        32.6
b.o. (inert)          85                           85                           85

#1. 141.8 left in main tanks excl. landing prop.
#1. 24.6 left in main tanks excl. land.,  plus 150 in payload tanks

The “dedicated tanker” configuration with extra tanks holding 150 tons in the forward section is the most efficient tanker of the three candidates.  It takes 6.3 of these to completely refill a crewed BFR vehicle in low Earth orbit,  using that 150 tons plus the excess in the main tanks beyond the landing budget.  The downside of this design is a third configuration to account fir in manufacturing.  It wouldn’t take much design refinement to eliminate the “.3” and get to 6 tanker flights. The target is 183-184 tons delivered to low Earth orbit.  The design (as crude as it is) delivers 175.

Provided that the crewed and cargo vehicles share the same inert masses,  then if flown at zero payload,  they arrive with enough excess propellant in the main tanks beyond the landing budget,  to enable complete refilling of a crewed vehicle in low Earth orbit,  if flown some 7.8 times.  This approach is less efficient from a number of flights standpoint,  but allows the advantage of having to account for only two configurations in the manufacturing effort.  It is probably not possible to get enough design refinement to eliminate the “.8” and get to 7 tanker flights.  8 flights is likely realistic.

The “real” numbers are going to be different,  once revealed by Spacex,  because these configurations will not all share the same inert weight as was assumed here.  The “flown-at-zero-payload” potential is close enough to the “dedicated tanker” potential that perhaps Spacex should investigate this possibility closer,  before “freezing” the crewed and cargo designs,  in spite of the inefficiency of flying with such large volumes of empty space on board.

Note also in the results figure that gee loads have been held to very tolerable values with very simple choices of engines and throttle settings.  In particular,  the landing settings have been chosen to enable either a 1-engine or 2-engine touchdown from the very same point in the descent trajectory.  You plan on flying as 2-engine,  but if one fails to ignite,  you just immediately double the thrust setting on the remaining engine.  The throttle margins are there to support that.