## Tuesday, September 11, 2018

### Velocity Requirements for Mars

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

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

Figure 1 – Min-Energy Hohmann Transfer Orbits to Mars

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

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

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

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

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

Arrival and Departure Speeds and Geometries

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

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

Figure 2 – Arrival and Departure Speeds for Hohmann Transfer

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

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

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

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

Departing From Earth

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

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

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

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

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

Figure 3 – Departing Earth From Either Orbit or the Surface

Arrival at Mars

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

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

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

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

Entry,  Descent,  and Landing (EDL)

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

Figure 4 – Mars Landings

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

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

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

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

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

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

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

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

Design Requirements Summary

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

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

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

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

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

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

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

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

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

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

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

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