Relevant Data for 2018 BFS Second Stage

Update 4-8-2024:  Should any readers want to learn how to do what I do (estimating performance of launch rockets or other space vehicles),   be aware that I have created a series of short courses in how to go about these analyses,  complete with effective tools for actually carrying it out.  These course materials are available for free from a drop box that can be accessed from the Mars Society’s “New Mars” forums,  located at http://newmars.com/forums/,  in the “Acheron labs” section,  “interplanetary transportation” topic,  and conversation thread titled “orbital mechanics class traditional”.  You may have scroll down past all the “sticky notes”. 

The first posting in that thread has a list of the classes available,  and these go far beyond just the two-body elementary orbital mechanics of ellipses.  There are the empirical corrections for losses to be covered,  approaches to use for estimating entry descent and landing on bodies with atmospheres,  and spreadsheet-based tools for estimating the performance of rocket engines and rocket vehicles.  The same thread has links to all the materials in the drop box. 

The New Mars forums would also welcome your participation.  Send an email to newmarsmember@gmail.com to find out how to join up.

A lot of the same information from those short courses is available scattered among the postings here.  There is a sort of “technical catalog” article that I try to main current.  It is titled “Lists of Some Articles by Topic Area”,  posted 21 October 2021.  There are categories for ramjet and closely-related,  aerothermodynamics and heat transfer,  rocket ballistics and rocket vehicle performance articles (of specific interest here),  asteroid defense articles,  space suits and atmospheres articles,  radiation hazard articles,  pulsejet articles,  articles about ethanol and ethanol blends in vehicles,  automotive care articles,  articles related to cactus eradication,  and articles related to towed decoys.  All of these are things that I really did. 

To access quickly any article on this site,  use the blog archive tool on the left.  All you need is the posting date and the title.  Click on the year,  then click on the month,  then click on the title if need be (such as if multiple articles were posted that month).  Visit the catalog article and just jot down those you want to go see.

Within any article,  you can see the figures enlarged,  by the expedient of just clicking on a figure.  You can scroll through all the figures at greatest resolution in an article that way,  although the figure numbers and titles are lacking.  There is an “X-out” top right that takes you right back to the article itself. 

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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

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. 

Payload and Cost-Effectiveness Comparisons to Mars

This article is one of 5 posted this date,  dealing with the abilities and the cost-effectiveness of 7 launch vehicles for delivering payloads to the surface of Mars.  4 of these articles provide the details for where the numbers came from,  1 provides the comparison among the all 7 launchers considered.  

This article is the comparison among the launchers. 

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

Here,  a few selected launch rockets are compared for payload delivery capability to the surface of Mars,  and the cost-effectiveness of doing so. Ref.1 shows how I arrived at these measures for the NASA SLS.  Ref. 2 shows how I arrived at these measures for the existing Spacex rockets Falcon-9 and Falcon-Heavy.  Ref. 3 shows how I arrived at these measures for the Spacex BFR system still early in development.  Ref. 4 shows how I arrived at an approximation to these measures for the Delta-IV Heavy and the Atlas V 551.

Most of these vehicles are launch rockets that contain a payload system within a protective shroud.  The shroud does not count against the thrown mass onto a min-energy Hohmann trajectory to Mars.  However,  there must be a custom payload adapter,  here assumed to be 10% of the thrown mass. That leaves a net payload mass.  The payload must have a means of landing,  which means part of the net payload mass is a lander spacecraft that can survive direct entry into the Martian atmosphere from the interplanetary trajectory,  plus it must touch down safely.  

At the payload numbers obtained here (all over a metric ton),  JPL's experiences with entry,  descent,  and landing (ref. 5) suggest that chutes will not be feasible,  so that a retro-propulsive landing must be attempted directly from the end of aerobraking hypersonics.  Such technologies are already flying (although JPL has never yet used them in this way),  while extendable or inflatable heat shields that reduce ballistic coefficient are not yet flying,  except very experimentally.  See Figure 1.

 Figure 1 -- Trajectories To and Onto Mars

Such a lander needs long-term storable propellants in order not to have very high inert weights and large excess propellants to make up boil-off losses.  Such a one-shot,  one-way lander would have a payload mass fraction (relative to lander ignition mass) of about 50%.  Ref. 6 shows why this is true.

One exception to this picture is Falcon-Heavy delivering payload to the surface via the now-cancelled Red Dragon concept.  Such has no payload adapter,  but the payload fraction of Red Dragon is low,  because the spacecraft is entirely too robust for the load it can land on Mars.  It really needs larger propellant tanks to be more attractive.  Because of the cancellation,  this issue is most likely moot. Dragon-related data are given in ref. 7.

The other exception is the Spacex BFR system,  which is still in the very early stages of development.  Only its Raptor liquid methane / liquid oxygen engines have had a lot of ground testing so far.  This concept carries its payload internally inside a second stage that is also the aerobraking-and-retropropulsion lander spacecraft.  The thrown payload masses need no correction.  But this system requires on-orbit refilling to leave low Earth orbit.  The best current estimate is 6 tanker flights to fully refill one interplanetary-bound second stage spacecraft.  That was found as part of ref. 8.

Bear in mind that the per-launch prices for SLS and BFR are nothing but best-guesses. The notation "m.ton" stands for metric ton,  and "MF" for the lander mass fraction payload/ignition.   Data:

So Of What Use Is All of This?

This is the last of a series of articles in which I estimated or determined the relevant numbers for several,  but by no means all,  possible launchers,  for the very specific mission of direct-entry landing payloads on Mars.  Anyone actually using these methods will need data for all potential launchers,  not just those few documented here. Similar methods using different numbers would apply to other destinations,  such as the moon,  or even various Earth orbits.

There are two things data like these can tell you.  First and most straightforward,  is selection of an existing launcher that can do the mission without paying too much for it. Second,  what launcher design concepts might be most worthy of future use or development? 

For the launcher selection problem,  the most relevant data are simply payload deliverable to the surface of Mars,  and the launch price you will pay for using it.  The best methodology is the simplest:  from the list of all possible launchers,  cross out all which cannot land on Mars the payload mass you want.  Then from among those remaining candidates,  look for the lowest launch price.

In practical terms,  you will need a reliability ranking as a second screening tool to cross out the unreliable vehicles.  And,  you will need to select one or two back-up candidates to pursue,  if your "best" candidate cannot be had.

The second problem of which designs deserve continued use or development,  would best be analyzed with the cost-effectiveness ratio,  which is a part of the data given here,  in the form of cost per delivered useful ton of payload.  That ratio is a cost/benefit ratio,  under the assumption of flying the vehicle at its full payload capacity.  The lowest cost per delivered useful ton is the "best" choice,  because as long as the payload tonnage will accomplish any mission at all, the lowest cost/benefit ratio does that mission the cheapest way.

If you don't fly full capacity,  you have to ratio-up the cost/benefit numbers:  flying at half the max payload system mass doubles the cost per delivered ton,  flying at one third of capacity triples the cost per delivered ton,  etc. This is the way to see technologies and design approaches that are getting too obsolete to be competitive in a market environment: too costly.  It is not constrained by either deliverable tonnage or reliability.

References:

1. Article dated Sept 2, 2018 titled "SLS Capabilities on Mars" posted here at http://exrocketman.blogspot.com

2. Article dated Sept 2, 2018 titled "Current Spacex Rockets to Mars" posted here at http://exrocketman.blogspot.com

3. Article dated Sept 2, 2018 titled "Future Spacex Rockets" posted here at http://exrocketman.blogspot.com

4. Article dated Sept 2, 2018 titled "Miscellaneous Rocket Data" posted here at http://exrocketman.blogspot.com

5. C.G. Justus and R.D. Braun, "Atmospheric Environments for Entry,  Descent,  and Landing (EDL)",  June 2007,  available online from NASA

6. Article dated Aug. 6, 2018 titled "Exploring Mars Lander Configurations" posted here at http://exrocketman.blogspot.com

7. Article dated March 6, 2017 titled "Reverse-Engineered Dragon Data" posted here at http://exrocketman.blogspot.com

8. Article dated April 17, 2018 titled "Reverse-Engineering the 2017 Version of the Spacex BFR" posted here at http://exrocketman.blogspot.com

Miscellaneous Rocket Data to Mars

This article is one of 5 posted this date,  dealing with the abilities and the cost-effectiveness of 7 launch vehicles for delivering payloads to the surface of Mars.  4 of these articles provide the details for where the numbers came from,  1 provides the comparison among the all 7 launchers considered.  

This article is a data sources and details article.

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

For the Delta-IV Heavy,  I found "8000 kg to TMI" per wikipedia as of 8-31-2018. This matches with the 8 metric tons for Mars missions shown for "existing launch vehicles" in the NASA Mission Planner's Guide for SLS (Reference 1).  The largest payloads among existing vehicles are carried by Delta-IV Heavy,  if one excludes Falcon-9 and Falcon-Heavy.  I also had a launch price for Delta-IV Heavy of $400M,  in reference 2. Here "TMI" means "trans-Mars injection",  presumably a min-energy Hohmann transfer orbit from Earth to Mars.

I found online a Popular Science magazine article.  I found it 8-31-2018.  It says "max 11,000 pounds to Mars for Atlas V",  without specifying the Atlas V configuration.  That 11,000 pound figure converts to 4.99 metric tons. The Atlas V user's Manual 2010 (available on-line) gave no guidance for missions outside Earth orbit. However,  that figure of just under 5 tons is fairly realistic for an Atlas V 500-series configuration,  so I used it here as "in the ballpark".  I had a launch price of $153M in reference 2 for an Atlas V 551 configuration.  I used this with the payload figure as an "in the ballpark" realistic estimate for a Mars launch with a 551 configuration.

Both vehicles fly their payloads within a shroud.  Per the methods used in reference 1,  I used the same 10% deduction from thrown mass for the payload adapter,  and the same 50% payload/ignition fraction for the direct-entry lander.  This presumes a direct entry from the interplanetary trajectory,  with a craft of fairly large ballistic coefficient,  so that chutes are infeasible because the altitudes are too low and timelines too short.  Under these circumstances,  retro-propulsion from the end of hypersonics (about 0.7 km/s) is required for a safe touchdown.  That retro-propulsion technology is flying operationally,  while extendible or inflatable heat shields that reduce ballistic coefficient are not yet flying,  except very experimentally.

Results follow:



vehicle................thrown mass...$M/launch..net payload..useful..$M/u.ton
............................to TMI m.ton.....................entry m.ton..m.ton... 
Delta-IV Heavy...8.000..............400.............7.20..............3.60.....111
Atlas-V 551.........4.99................153.............4.491............2.245...68



Note that m.ton means metric ton,  net payload is the mass at entry and at ignition for the lander,  and 50% of that lander mass is the useful payload delivered to the surface of Mars. Note also that these numbers are inherently imprecise,  but not by factors of 2 or more.

References:

1. 2014 Mission Planner's Guide released by NASA,  available on-line as file SLS_Mission_Planners_Guide_Ver1_Aug2014.pdf

2. Article dated Feb 9, 2018 titled "Launch Costs Comparison 2018" posted here at http://exrocketman.blogspot.com           

Future Spacex Rockets

This article is one of 5 posted this date,  dealing with the abilities and the cost-effectiveness of 7 launch vehicles for delivering payloads to the surface of Mars.  4 of these articles provide the details for where the numbers came from,  1 provides the comparison among the all 7 launchers considered.  

This article is a data sources and details article.

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

There is only one system publicly revealed right now:  the so-called "BFR",  commonly referred to as "Big Falcon Rocket".  This was first revealed in 2016 at a meeting in Guadalajara.  As first revealed,  the two-stage system was larger than depicted today.  In a 2017 presentation,  a smaller version 9 m in diameter was revealed,  and characterized in more detail.  There are some differences between the 2017 presentation still on Spacex's website,  and a talk given at the Mars Society Convention in 2018,  but these seem fairly minor.  The 2017 version was reverse-engineered for realistic performance capabilities estimates in Reference 1.  These largely confirm what Spacex has so far revealed. Some people refer to the first stage as "BFR",  and to the second stage as "BFS",  for "Big Falcon Spaceship".

The first stage has 31 Raptor liquid methane / liquid oxygen engines in a configuration that looks like a scaled-up Falcon first stage core,  complete with grid fins and landing legs.  It is very clear that this is intended to fly back to the launch site,  and land on the launch pad,  almost exactly the same way as Falcon-9 and Falcon-Heavy first stages recover today.  It must retain a modest propellant allowance for that return and landing,  as documented in ref. 1.

The second stage is a lifting-body with an external heat shield,  intended for aerobraking flyback,  followed by a tail-first retro-propulsive final touchdown.  This recovery of the second stage for re-use is new.  The 2017 presentation shows 6 engines:  4 vacuum types,  2 sea-level types,  all liquid methane / liquid oxygen Raptor engines.  The 2018 Mars Society talk says there are 7 engines,  but did not give details.  Payload was 150 tons in 2017,  is now "over 100 tons".

For comparative purposes here,  the reverse-engineered 2017 version documented in ref. 1 will be used.  Nominal max payload is 150 metric tons,  carried entirely internally,  much the same as an airplane or a ship would do.  There is no payload adapter or custom delivery spacecraft in this design.  The second stage literally is the lander.

Key to understanding this vehicle design concept is the propellant allowance for landing,  after returning from low Earth orbit.  This estimate was documented in ref.1,  and must still be aboard at entry,  or else the vehicle will crash.  This is not a large allowance,  the vehicle burns most of its propellants (some 1100 metric tons in total) accelerating from the stage point to low Earth orbit.  It needs no refilling in low Earth orbit if used as a transport to orbit.  There is plenty of propellant aboard the two stages to accomplish flying to orbit,  and flying home again separately. Thus to orbit,  that is one launch,  one payload delivery.

Multiple Configurations

Because of the generality of the internal payload carriage,  and the great size of the vehicle,  the same system may be used for missions outside of Earth orbit,  given refilling with propellants on-orbit.  Thus,  there are probably at 3 versions of the second stage design:  a passenger-cargo vehicle,  an unmanned all-cargo vehicle,  and a tanker for refilling on-orbit.  The passenger-cargo version was well-covered by the 2017 presentation,  and some external views shown of the unmanned all-cargo version with a giant clamshell door.  No details of the tanker have yet been published.  According to Spacex,  all three share the same engine and propellant tankage sections.  Only the forward sections differ.   This is a design concept approach that makes good engineering sense.

Spacex claims that this two stage system,  with refilling on-orbit,  can take 150 ton payloads to the moon and to Mars.  Payloads on the return voyages are restricted to about 50 tons,  they say.  These claims are verified as feasible by the reverse-engineering analyses in ref. 1.  Spacex has not defined how many tanker flights are required for a full-capacity refill on-orbit,  nor have they said how much refill propellant these tankers can carry.  On the assumption that each tanker can carry 150 tons,  or perhaps a just little bit more,  then 6 tanker flights plus what would have been the landing allowance,  should be pretty close to the 1100 tons capacity.  So for a moon or Mars flight,  a total of 7 launches should put 150 tons delivered useful payload mass on the surface of the moon or Mars.

Missions to the Moon

Spacex claims that,  with reduced return payload and no refilling on the moon,  the vehicle can escape the moon and return home to an aerobrake/retro-propulsive landing on Earth.  The data in ref.1 confirm the feasibility of doing that mission without refilling on the moon.  There is insufficient propellant capacity to recover in Earth orbit.   The aerobrake entry is the main means of decelerating at Earth,  with retro-propulsive touchdown. If realized in practice,  this system could bring a permanently-manned base or presence on the surface of the moon within easy reach.  This would be 7 launches to bring 150 delivered useful tons per landing on the moon,  plus the vehicle is large and spacious enough to serve as habitable space while there.

Missions to Mars

It is fairly common knowledge that missions to Mars are far more demanding than missions to the moon.  A refilled-on-orbit second stage passenger-cargo vehicle (with full 150 ton payload) will expend most of its propellant load just getting from Earth orbit onto the min-energy interplanetary trajectory to Mars,  called a Hohmann transfer orbit. It will need the remainder to land on Mars after aerobraking entry,  direct from the interplanetary trajectory. There is no propellant to stop in Mars orbit,  the touchdown allowance (~1 km/s dV) is simply insufficient for that (>1.6 km/s dV).

The calculations of ref. 1 confirm feasibility of these Spacex claims.  Without refilling on Mars,  the vehicle is stranded there,  forever.  To fly a faster trajectory than Hohmann min energy requires carrying a smaller payload,  though.  Ref. 1 found little or no such excess capability for faster trips at full payload.

Given the infrastructure to create propellant-quality liquid methane and liquid oxygen on Mars,  in sufficient quantities within reasonable times,  the vehicle could then be fully refilled with the full 1100 ton capacity.  Reducing payload to 50 tons,  the vehicle then has the capacity to escape from Mars directly onto the min-energy interplanetary trajectory home. It cannot stop in Earth orbit,  it only has the propellant to make a free-return aerobrake entry,  followed by the final retro-propulsive touchdown. 

These vehicles are large and spacious enough to serve as habitation spaces while on Mars.  As with the moon missions,  it is 7 launches total to send one vehicle to Mars,  and deliver 150 tons of useful payload to the surface. They are stranded there permanently,  unless propellants can be produced on Mars to refill them.

Cost-Effectiveness for Comparison

This depends upon the per-launch price of a BFR launch.  Such is not yet available.  Costs are nothing but wild guesses at this time.  The published price of a Falcon-Heavy is $90 million.  BFR is about twice the size of Falcon-Heavy,  but will be fully reusable,  not just first-stage reusable.  As a wild guess,  call it $100 - 200 million per launch. That should be somewhere close,  if not in,  the ballpark.

We do not need to adjust the published payload numbers,  because of the internal carriage inside a vehicle that is also the lander.  We need no payload adapter deduction,  and we need no lander mass fraction effect.  That results in the following:

mission pay.m.ton.launches....$M/launch.cost,$M...$M/del.ton
LEO......150..........1................100-200.....100-200...0.67-1.33
moon.....150..........7(6 tanker).100-200....700-1400.4.67-9.33
Mars......150..........7(6 tanker).100-200....700-1400.4.67-9.33

References

1. Article dated April 17, 2018,  titled "Reverse-Engineering the 2017 Version of the Spacex BFR",  located on this site http://exrocketman.blogspot.com