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

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.

**. 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***For orbital departure from Earth, that is dVorb = 3.85 km/s, or an achieved Vdep = 11.75 km/s from the surface***C3 values are also shown for both Earth and Mars.***. To fly anytime, you must design for an orbital entry burn of dVorb = 1.80 km/s.***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.

**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.

**There is no way around that.***It will take thrusting at 3-4 standard Earth gees to zero that speed before touchdown.*
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,

**. This jigger factor recommendation is also given in Figure 4.***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
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 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.

This is incredible work. It is detailed enough to be a complete reference for anyone interested in missions to Mars.

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Thanks you, sir! -- GW

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