This op-ed page cartoon appeared in the Saturday Waco "Tribune-Herald" this date. It was embedded in an opinion piece about which people will argue. But this cartoon itself is actually funny in a black humor sort of way, and is actually rather truthful about what has been going on.

Here it is without further comment. Enjoy!

## Saturday, January 18, 2020

## Thursday, January 2, 2020

### On High Speed Aerodynamics and Heat Transfer

These topics are complicated, interconnected, and difficult to master well enough to enable
doing the work. Yet they can be intuitively
understood easier than most people suspect.
This is really just an “understanding” article, not a “how-to” article.

For atmospheric flight from high subsonic to high
supersonic, and even low
hypersonic, the standard compressible
flow analyses apply, which are based
upon the

__ideal gas assumption__. Primarily, that means all the kinetic energy of motion goes into internal energy, which raises temperature. None goes into ionization. And, the usual equation of state PV = nRT can be used.
This energy deposition effect shows up quite directly as the
“stagnation” or “total” temperature trend with Mach number, for any given static (thermodynamic)
temperature, and for any given gas
specific heat ratio ϒ. That equation is
quite simple:

T

_{tot}= T_{static}(1 + c1 M^{2}) where c1 = 0.5*(ϒ – 1)
That same ideal-gas fluid mechanics model underlies much of
the heat transfer correlations that apply to various geometries and situations. At low speeds, even fairly high subsonic, there is not enough viscous dissipation due
to velocity shear in the boundary layer, to raise its cross-layer temperature
profile. Thus, under those conditions, the static gas temperature is usually taken
as the driving temperature for heat transfer to the adjacent surface.

Once supersonic, that
boundary layer temperature profile shows a temperature peak termed “recovery
temperature”, which is only a little
less than the total temperature. Because
that hotter layer of gas is so closely adjacent to the surface (being down in
the boundary layer), it effectively
drives the heat transfer to that surface. In that case,
the driving temperature is the recovery temperature.

The equation for calculating recovery temperature is also
fairly simple, but requires knowledge of
the gas Prandtl number PR and whether or not the flow is laminar or turbulent:

T

_{rec}= r*(T_{tot}-T_{static}) + T_{static}where r = PR^{n}and n = 0.5 laminar, 0.333 turbulent
In the supersonic speed ranges where those correlations
actually apply, there are heat transfer
models for stagnation zones that use total temperature, and there are heat transfer models for
lateral surfaces that use recovery temperature.
Those lateral surface models can be the simpler static temperature
models, but only if the speed is slow
enough that no viscous dissipation occurs (essentially subsonic).

The “division” between “supersonic flow” and “hypersonic
flow” has to do with how rapidly the overall shock wave geometry changes with
increasing speed. That rate of change is
quite rapid at low supersonic speeds,
but decreases to near-zero at hypersonic speeds. The changeover point depends upon the
bluntness of the flying object. Very
blunt objects such as space capsules traveling heat-shield-forward are
hypersonic at about Mach 3, while “pointy”
objects like aircraft at near-zero angle-of-attack, or missiles, are hypersonic at about Mach 5.

The limit to this type of compressible flow analysis occurs
at hypersonic speeds when the gas begins to ionize, throwing the underlying ideal gas-based
compressible fluid mechanics model into error.
That in turn throws the heat transfer correlations into error. Once the error becomes significant, one needs to be using non-ideal gas
assumptions, where the equation of state
PV = nRT no longer applies. And, one needs to be using heat transfer
correlations actually developed for that regime.

There is a rule-of-thumb for the effective gas temperature
that drives heat transfer, in such
hypersonic flight that ionization is significant. That rule-of-thumb is only approximate, but quite simple:

Temperature in deg K is
numerically equal to velocity in meters/second

What Figure 1 shows is the trends of ideal-gas compressible
flow-based total temperature, and the
entry rule-of-thumb temperature, versus
flight Mach number from 0 to 10, for air
at 15 C.

Figure 1 – Calculated Driving Temperature Trends Mach 0 to 10

In the figure, the
blunt and “pointy” speeds for hypersonic flight are noted. The two temperature estimates cross at Mach
5, and are still close at Mach 6, but not so much at Mach 7. Your vehicle aerodynamic coefficients (such
as drag and lift coefficients) become more-or-less constant at the hypersonic
transition points indicated.

Once you are supersonic,
the heat transfer correlations that use total or recovery temperature
apply. These and the compressible fluid
mechanics techniques are the models to use,
up to about Mach 5 for sure,
maybe to Mach 6, but probably not
Mach 7. The errors of not allowing for
ionization are getting too large for any sort of design analysis that you could
trust, past about Mach 6.

This not only applies to aerodynamics and heat
transfer, but also to any sort of
propulsion cycle analysis that uses ideal-gas compressible fluid
mechanics! Or to anything else that
uses those same compressible fluid mechanics modeling techniques.

Figure 2 shows just how rapidly the modeling error increases
as speeds increase to those in the entry range.
This is the same sort of plot of the same variables as in Figure 1, just extended out to Mach 30. Typical of a re-entering warhead would be
Mach 15.

At Mach 15, the entry
rule-of-thumb is that heat transfer driving temperature for the heat shield
stagnation point will be near 5000 K,
and actual non-ideal gas models would get something in that same
ballpark. But classical ideal-gas
compressible flow models would estimate that temperature to be about 13,000
K. The error disparity is immense! The difference is due to kinetic energy going
into ionization instead of raising thermodynamic temperature. It’s no longer the same gas.

**Recommendations:**

If you are very “pointy” in shape, the supersonic/hypersonic transition is about
Mach 5; if blunt, it is just about Mach 3.

“Pointy” objects at very high angles of attack (~20-30
degrees or more) are going to behave more like blunt objects, because the “pointy” nose simply does not
face directly into the oncoming slipstream.

You may use ideal-gas based compressible fluid mechanics up
to about Mach 5, to maybe Mach 6, without significant error. For flight in air, that corresponds to roughly 2500 K total
temperatures, just about where air
dissociation into ionized plasma becomes significant.

Beyond about Mach 6,
you need to be using a non-ideal gas based fluid mechanics model, or else just real experimental data. None of
the compressible-fluids-based heat transfer correlations apply.

Once there is supersonic flow, your heat transfer models for lateral
surfaces should include viscous dissipation effects: they should be formulated in terms of
recovery temperature as the driving temperature. It is the hot sublayer in the boundary layer
whose temperature drives heat transfer.

Compressibility itself is modeled differently for heat
transfer: typically as a reference
temperature for gas properties termed T* that is distinct from the average film
temperature. One needs to be doing that,
even at high subsonic speeds.

Beyond about Mach 6,
your stagnation point heat transfer model should switch to an
entry-range correlation; below
that, you can use one based on classical
compressible flow total temperatures.

If you are flying fast enough that ideal-gas compressible
fluid mechanics is not applicable, it is
very likely that any propulsion cycle analysis (other than a simple rocket) is
also inapplicable, as these are almost
universally based on ideal-gas compressible fluid flow.

**Heat Protection Applications:**

For entry hypersonics,
the situations depicted in Figure 3 apply for a blunt capsule traveling
above Mach 3, at zero or low
angle-of-attack (AOA). There is a
stagnation point near the center of the heat shield. That is the location of highest applied
pressure, and highest heat transfer
coefficient “h”.

Because of the bluntness,
subsonic flow behind the bow shock prevails over the heat shield, with the sonic line near its periphery, as illustrated. Most such shapes have a “tumble-home” angle
exceeding 20 degrees, so that the flow
separation point is also at the periphery of the heat shield.

The same tumble-home angle limits the achievable AOA to no
more than the tumble-home angle, because
otherwise, the flow would re-attach on
the more-windward side, leading directly
to very much higher heat transfer coefficient “h”, due to the scrubbing action.

Otherwise, the
lateral sides of the capsule are within the wake zone, where velocities relative to the surface are
quite low, and so there is little
scrubbing action. That means heat
transfer coefficients “h” are low,
relative to those on the other surfaces.
These lateral sides therefore receive the least heating rates, far below any of those seen on the heat
shield surface. They could cool by
re-radiation, if the local equilibrium material
temperature is low enough to be tolerable for the material.

The situation for “pointy” shapes at high angle-of-attack is
actually rather similar, as illustrated
in Figure 4. If dead broadside, there is a stagnation line along the belly. If otherwise high AOA, there is a stagnation point near the nose end
of the belly, with near-stagnation
conditions along what otherwise would have been the stagnation line along the
belly.

Either way, it is the
crossflow picture that is informative.
While the effective cross section shape varies with AOA, the basic conditions are otherwise similar, as illustrated in the figure. The sonic line delimiting the subsonic region
is much closer to the stagnation point or line,
with supersonic flow over much of the windward side. The resulting heat transfer coefficients are
higher than one might otherwise expect, due
to the larger supersonic scrubbing action,
in spite of the lower pressures.

Figure 4 – Typical Flow Field Characteristics for a “Pointy”
Shape at High AOA in Entry Hypersonics

The point at which flow separation typically occurs in
turbulent flow is at (or just past) the maximum cross section width, as shown.
Conditions in the separated wake zone are very low velocity, with low scrubbing action, and low heat transfer coefficients.

If flow were instead laminar, the separation point would occur somewhat
upstream of the location of maximum width.
However, that would obtain only
on very small objects. Otherwise, the conditions are pretty much similar to
what is pictured in the figure.

Bear in mind that hypersonic heating is both convective and
radiative. The convective heating varies
with velocity cubed, and dominates the
picture at speeds below 10 km/s in air.
Radiative heating comes from the incandescent plasma that is the shocked
flow surrounding the object. This effect
varies as velocity to the 6

^{th}power, and dominates the picture above 10 km/s in air.
Even if the atmospheric gas is not Earthly air, there is not a lot of difference, qualitatively. Specific heat ratio and Prandtl number are
different, so the numbers are a little different.

On the windward side,
there must be some sort of ablative or refractive heat shield
material, in direct contact with a
significant mass of structure within the object. That structure is the heat sink, into which the heat conducted inward must be
deposited. The whole process is a short
transient, only minutes in
duration.

*And THAT is why entry heat protection is fundamentally a transient heat-sinking process quite distinct from sustained hypersonic atmospheric flight!*

Because of the far lower heating, the lateral-side heat protection problem is
quite different, as illustrated in
Figure 5. If the entry situation is not
too demanding, and the lateral wall
material can survive the high equilibrium temperatures, then bare metal sides can be used, as in the old Mercury and Gemini capsules
returning from low Earth orbit at about 8 km/s speeds.

If the situation is a little more demanding, similar to Apollo returning from the moon at
just under 11 km/s, there must be some
heat shield ablative or refractory over the structure of the lateral
sides. This is also shown in the
figure. That substructure is the heat
sink for the lateral heat shield material.

This kind of “backshell” construction (whether it needs heat
shielding or not) is required in order to protect cargo within that is
“delicate”, in the sense that its
temperature may not be allowed to go high enough to re-radiate. Such is typically in the vicinity of 1000+ F
= 811+ K for steels and other alloys.

On the other hand,
the cargo might be “tough”, in
the sense that it can withstand getting hot enough to re-radiate effectively
all by itself. In that case, there is no need for any backshell at
all, only a windward-side heat shield is
required.

For re-radiation to work as the only cooling, then the material temperature and
spectrally-averaged emissivity must be high enough for that re-radiation to
effectively balance all of the convective and radiative heat inputs at their
peak values, and to do this at an
acceptable material temperature.

Figure 5 – Technical Solutions For Heat Protection During
Entry

For refractory heat shields,
the surface temperature may freely “float” to equilibrium. Only practical material service temperatures
limit this. Otherwise, the balance is the same as for ablative heat
shields, except that there is no energy
consumed by pyrolysis.

Minimum conduction inward occurs when the heat sink
temperature has maximized. The
temperature rise of the heat sink, its specific
heat capacity, and the mass of the heat-sink substrate, must together be enough to contain all of the
heat conducted into it, during the total
entry heating transient.

**Steady Low-Hypersonic Flight:**

For analyzing steady cruise up to about Mach 6 in the
atmosphere, the heat protection problem
tends to become a steady-state issue,
not a transient heat-sink issue.
This is because the time scale is so much longer: several minutes to a few hours.

The same basic ideal-gas compressible-flow analyses
apply, except that your heat sink “gets
full” quicker than you can reach your destination.

__That means there are two, and only two, practical means to deal steady-state with the heat energy you continuously absorb__:
(1)
Re-radiate the energy away to the environment

(2)
Put the absorbed energy into the propulsive
fuel, so that it ultimately exits the
tailpipe

To radiate away the energy means the radiating surface must
get hot, it must have good
spectrally-averaged thermal emissivity to be efficient, and it must have a direct view of the
environment. The Earthly environment has
a warm temperature, which reduces the
energy radiation rate somewhat. A form
of Boltzmann’s Law applies to the re-radiated energy rate:

Q/A, BTU/hr-ft

^{2}= e σ (T^{4}– T_{E}^{4}) for T’s in deg R and σ = 0.1714 x 10^{-8}BTU/hr-ft^{2}-R^{4}^{}

where T is the material temperature, T

_{E}is the Earthly environment temperature (near 540 R = 300 K), e is the spectrally-averaged emissivity (a number between 0 and 1), and σ is Boltzmann’s constant.
If instead you “dump” the heat into the fuel as it is used, it will get very hot, and

__you must prevent that fuel from boiling__, which produces vapor lock, that stops the propulsion. That requires very high fuel delivery pressures, especially in the passages where it is the coolant for the aeroheated part.
Depending upon the nature of the fuel, there is also the risk of “coking”
deposits, which will plug up small
passages very quickly, even if boiling
is successfully avoided.

Above Mach 6, the
compressible flow and heat transfer relations have to be replaced by their
non-ideal gas equivalents, but
Boltzmann’s Law still applies for re-radiation. It’s still the same
steady-state heat balance for each and every part of the vehicle.

**Airbreathing Engine Cycle Analysis Applications:**

Gas turbine engines of multiple types can be modeled fairly
accurately with simple “pressure ratio” models,
because the compressor pressure-rise and turbine pressure-drop dominate
the cycle pressure picture by far, over
all the other effects. Those “others” would
be the inlet pressure rise, and the
pressure losses associated with all the other components. These are usually input as fixed ratios, because the variations with operating
conditions are quite small compared to compressor and turbine pressure-rise and
pressure-drop effects.

That last becomes increasingly inaccurate as flight speeds
get hypersonic, because the variations
of component performance with operating conditions are going to get far more
important as conditions get more extreme.
Offsetting that is

__this simple fact__: so far, there are no turbine engines that have ever been operational, which flew any faster than about Mach 3.5.
Those same pressure ratio models can be used to model
subsonic combustion ramjet engines,

__but this is far too imprecise to be a useful design analysis!__There are__no__compressor pressure-rise and turbine pressure-drop phenomena in a ramjet. The only pressure-rise item is the inlet, and its variation with conditions and its interplay with the pressure drops of all the other components, dominates ramjet behavior. These things simply__cannot accurately be modeled__as “typical constant ratios”.
At the other end of the spectrum is finite-difference
computer fluid dynamics modeling. This
is accurate to the extent that the turbulence model, the combustion models, the flow separation models, and the ionization/non-ideal gas models are
all accurate (not all codes have such).
Such analyses require great effort to set up and to interpret the
results. One analysis for one situation in
one design means the investigator must run a lot of them, for just the one design. However,
this is about the only realistic way to model supersonic combustion
ramjets (scramjets), especially at
speeds well above Mach 6.

In between these two modeling extremes are the quasi-1-D ideal-gas
compressible flow-based models of subsonic combustion ramjet cycle
analysis. These are quite accurate, up to the Mach 6 point where the ideal gas
model fails, and they require a lot less
effort to set up, and very little to
interpret.

These can be tailored to provide sizing or performance, and for repeated performance runs across the
flight envelope. They provide fluid flow
state information at every modeled location within the engine. Those are all very distinct advantages.

__But they inherently do not provide accurate results past about Mach 6.__**Recommended Simple Heat Transfer Models for Back-of-the-Envelope Stuff:**

These divide into what applies in when compressible fluid
mechanics is “good”, and what applies
when it is “not good”. That changeover
is about Mach 6.

*Supersonic to About Mach 6*

These take the forms of lateral surfaces, stagnation zones, and internal duct flow (for propulsion items
such as combustors and subsonic ducts).
You look up gas properties from standard tables, or else estimate them with empirical
equations.

*Turbulent flow with compressibility and viscous dissipation on a flat plate parallel to freestream; applicable to exposed skins in high speed flight at speed V*

Need: total T

_{t}, and static T, plus fluid Prandtl number Pr and plate surface T_{s}
plus
density rho and velocity V at edge of boundary layer, and length L

Calculations: recovery
factor r = Pr

^{1/3}
recovery
temperature T

_{r}= r (T_{t}– T) + T
ref
temp. T* = 0.5(T + T

_{s}) + 0.22(T_{r}– T) this models compressibility
evaluate
properties at T*

ReL*
= rho V L / mu (properties at T*, V = freestream/edge of boundary layer)

average
NuL* = .036 ReL*

^{0.8}Pr^{1/3}
average
h = NuL* k/L (k at T*)

Q/A
= h(T

_{r}– T_{s}) this models the viscous dissipation, positive to surface for T_{r}>T_{s}
Source: Chapman
(ref. 2) eqn. 8.41, attributed therein
to Eckert

*Stagnation-Point heating in very high speed flow (supersonic and hypersonic)*
Need: stagnation T

_{t2}, P_{t2}, k_{t2}, mu_{t2}, Pr_{t2}, and rho_{t2}behind the shock wave, surface T_{s}, freestream V and rho; diameter D = 2 R_{n}, where R_{n}is the nose radius
Calculations: Rpt
= P

_{t2}/P_{t1}= (N1/D1)^{E1}*(N2/D2)^{E2}where
N1
= (γ + 1)M

^{2}, D1 = (γ – 1)M^{2}+ 2, and E1 = γ/(γ – 1)
N2
= γ + 1, D2 = 2 γ M

^{2}– (γ – 1), and E2 = 1/(γ – 1)
P

_{t2}= P_{t1}Rpt (this procedure is total pressure ratio across a normal shock, any γ)
ReD = rho

_{t2}V D / mu_{t2}
Cylinder: NuD = 0.95 ReD

^{1/2}Pr_{t2}^{0.4}(rho/rho_{t2})^{1/4}applies to leading edges
Sphere: NuD = 1.28 ReD

^{1/2}Pr_{t2}^{0.4}(rho/rho_{t2})^{1/4}applies to nose tips
h
= NuD k

_{t2}/ D
Q/A
= h(T

_{t1}– T_{s}) where T_{t1}= total ahead of wave = T_{t2}behind wave
Source: Chapman
(ref. 2) eqn. 8.45, see also ref. 1 for
total pressure ratio across shock wave

*Turbulent flow inside a pipe or duct or tube, with a nontrivial difference between fluid T and surface T*_{s}; applicable to combustor and subsonic inlet air duct inside film coefficients
Need: flow
rate w, duct diameter D, fluid static T, and surface T

_{s}
Calculations: evaluate
all properties at bulk T fluid, plus a second mu

_{s}at T_{s}
ReD
= rho V D / mu = 4 w / pi D mu (second
form very convenient!)

NuD
= 0.027 ReD

^{0.8}Pr^{1/3}(mu/mu_{s})^{0.14}(“Seider and Tate”)
h
= NuD k/ D

Q/A
= h(T – T

_{s}) positive to surface for T > T_{s}
Source: Chapman
(ref. 2) eqn. 8.16, attributed therein
to Seider and Tate

*Above About Mach 6*

__This takes the form only of stagnation heating__. The lower values farther away from the stagnation zone must come from experimental data for each shape and situation. However, for a blunt capsule heat shield, a first-cut over-design is to size the heat shield to the stagnation zone requirements. It must be “that thick” there, and can be about half that thick near the sonic line.

The stagnation point heating model is proportional to
density/nose radius to the 0.5 power,
and proportional to velocity to the 3.0 power. The equation used here is H. Julian Allen’s simplest
empirical model from the early 1950’s,
converted to metric units. It is:

q, W/sq.cm = 1.75 E-08 (rho/rn)^0.5
(1000*V)^3.0, where rho is kg/cu.m, rn is m,
and V is km/s. The 1000 factor
converts velocity to m/s.

The source for this (that I have in my possession) is a US
customary units form of the same equation,
obtained from ref. 3.

**Getting Gas Properties**

If you cannot obtain more accurate data from tabulations in various
references, there are some empirical
estimating relations that will get you “into the ballpark”. These are from the “grab bag” chapter of my
yet-to-be-published book on ramjet propulsion,
ref. 4.

It is always preferable to look up the properties of real
gases and liquids in standard references.
For hot combustion gases, this is
often problematical. However, it is possible to approximate the properties
from simple inputs. I obtained these
correlations informally from a colleague,
James M. Cunningham, who was the
head thermal analyst at Rocketdyne/Hercules in McGregor, Texas, about 2 to 4 decades ago. These also work reasonably well for air
itself, although real data tables for air
are better. Choose a gas MW and γ
appropriate to the temperature range you are working in. Those and the temperature are all you need to
estimate realistic properties for heat transfer purposes. Ideal gas behavior is inherently assumed.

*Input items as constants (not very dependent upon temperature at all):*

Gas molecular weight MW (can
be summed up from the chemical balance equation)

Gas specific heat ratio γ (usually
in the vicinity of 1.2 for real combustion-product gases)

*Functions of inputs but not very dependent upon temperature:*

Specific heat at constant pressure c

_{p}= 1.987 γ / [(γ – 1) MW] units will be BTU/lbm-R
Prandtl number Pr = 4 γ / (9γ – 5) dimensionless

Gas constant R,
ft-lb/lbm-R = Ru/MW where
Ru = 1545.4 ft-lb/lbmole-R

*Functions of inputs and strongly-dependent upon temperature:*

Viscosity µ,
lbm/in-sec = 46.6 x 10

^{-10}MW^{0.5}(T, R)^{0.6}or µ, lbm/ft-sec = 5.592 x 10^{-8}MW^{0.5}(T, R)^{0.6}
Thermal conductivity k, BTU/hr-ft-R = [1 x 10

^{-4}(9γ – 5) (T, R)^{0.6}] / [(γ – 1) MW^{0.5}]

*General Remarks:*

None of these above are considered to be significantly
pressure-dependent at all. The property
that is strongly pressure-dependent is density.
The ideal-gas relation to define density is:

rho, lbm/ft

^{3}= (P, psfa) / [(R, ft-lb/lbm-R)*(T, R)] (“psfa” is lb/ft^{2}, absolute)
If your situation is not quite amenable to the ideal gas
model, you can still use P = Z rho R
T, where Z is an empirical function of
P, T, and the critical P

_{crit}and T_{crit}for your gas. Reference 5 is an old thermodynamic textbook that contains a universal approximation chart for Z. Any modern text should contain the same chart. This is more applicable to cold conditions approaching the triple point.
It should be noted that the usual Mach number formulation of
compressible flow analysis does in fact assume both adiabatic and ideal-gas
behavior!

**References:**

There are many references for classical compressible fluid
mechanics. But fundamentally, virtually all of the basics in any of
them, trace back to the famous old NACA
report number 1135:

#1. NACA Report 1135, “Equations,
Tables, and Charts For
Compressible Flow”, Ames Research Staff, 1953.

There are many books available on heat transfer, covering a variety of situations with
empirical correlations for each. My
reference is an old college textbook on the subject:

#2. Alan
J. Chapman, “Heat Transfer” second
edition, MacMillan, 1967.

There is a variety of good information on very many topics in
the following reference. I cite it for
the simple entry stagnation heating equation attributable to H. Julian Allen, equation 4B-4 page 520. In the reference, this is given in US customary units. I converted this to metric, specifically for entry calculations using
speed in km/s and heating rates in W/cm

^{2}.
#3. “SAE Aerospace Applied
Thermodynamics Manual”, second
edition, Society of Automotive
Engineers, 1969.

The next reference is not yet available, but soon should be. AIAA has chosen not to offer it, so that I must self-publish it.

#4. Gary W. Johnson, “A Practical Guide to Ramjet
Propulsion”, yet to be published but copyrighted as of 2017.

This one is just another old textbook on classical (not
statistical) thermodynamics. Any modern
classical thermodynamics textbook should have the same information in it.

#5. Gordon J. Van Wylen and Richard
E. Sonntag, “Fundamentals of Classical
Thermodynamics”, John Wiley and
Sons, 1965.

Labels:
airplanes,
launch,
ramjet,
space program

## Saturday, December 7, 2019

### Analysis of Space Mission Sensitivity to Assumptions

For any given vehicle design, what one assumes for mission delta-vees, vehicle weight statements, course corrections, and landing burn requirements greatly affects
the payload that can be carried. The
effect is exponential: variation in
required mass ratio with changes in delta-vee and exhaust velocity.

This analysis looks at trips from low Earth orbit to direct
entry at Mars, and for the return, a direct launch from Mars to a direct entry
at Earth. The scope is min-energy
Hohmann transfer plus 3 faster trajectories (see ref. 1).

The vehicle under analysis is the 2019 version of the Spacex
“Starship” design, as described in ref.
2. The most significant items about that
vehicle model are the inert mass and the maximum propellant load. For this study, the vehicle is presumed fully loaded with
propellant at Earth departure, and at
Mars departure. See also Figure 1. Evaporative losses are ignored.

Figure 1 – Summary of Pertinent Data for 2019 Version of Spacex “Starship” Design

Since a prototype has yet to fly, the design target inert mass of 120 metric
tons is presumed as baseline.
Uncertainty demands that inert mass growth be investigated. To that end,
the average of that design target and the 200 metric ton inert mass of
the so-called “Mark 1 prototype” (that average is some 160 metric tons) is used
to explore that effect.

As currently proposed,
the vehicle has six engines.
Three are the sea level version of the “Raptor” engine design, and the other three are vacuum versions of
the same engine design (basically just a larger expansion bell). I have already reverse-engineered
fairly-realistic performance for these in ref. 3. Because of the smaller bells, the sea level engines gimbal significantly, while the vacuum engines cannot.

*Thus it is the sea level engines that must be used to land on Mars as well as Earth: gimballing is required for vehicle attitude control.***Analysis Process**

As shown in Figure 2,
the analysis process is not a simple single-operation calculation. The vehicle model provides a weight statement
and engine performance. The mission has
delta-vee requirements for departure,
course correction, and
landing, which must be appropriated
factored (in order to get mass ratio-effective values). There are two sets of analysis: the outbound leg from Earth to Mars, and the return leg from Mars to Earth.

Each leg analyzes 3 burns.
Earth departure, and course
correction are done with the vacuum “Raptor” engines, while the landing on Mars is done with the
sea level “Raptors” to obtain the necessary gimballing. Mars departure and course correction are done
with the vacuum “Raptor” engines (Mars atmospheric pressure is essentially
vacuum). The Earth landing is done with
the sea level “Raptors” to get the gimballing and to get the atmospheric
backpressure capability.

This analysis is best done in a spreadsheet, which then responds instantly to changes in one
of the constants (like an inert mass or a delta-vee). That is what I did here.

Referring again to Figure 2,
for each burn, there is an
appropriate vehicle ignition mass. At
departure, it is the ignition mass from
the weight statement. For each subsequent
burn, it is the previous burn’s burnout
mass. Each burn’s burnout mass is its
ignition mass divided by the required mass ratio for that burn, in turn figured from that burn’s delta vee
and the appropriate exhaust velocity.

For each burn, the
change in vehicle mass from ignition to burnout is the propellant mass used for
that burn. For the first burn, the propellant remaining (after the burn) is
the initial propellant load minus the propellant mass used for that burn. For the subsequent burns, propellant remaining is the previous value of
propellant remaining, minus the
propellant used for that burn.

After the final burn,
the propellant remaining

__cannot__be a negative number! If it is, one reduces the payload number originally input, and does all the calculations again. If this done in a spreadsheet, this update is automatic! Ideally, the propellant remaining should be exactly zero, but for estimating purposes here, a small positive fraction of a ton (out of 1200 tons) is “close enough”.**This particular input (payload mass) is revised iteratively until the final burn’s remaining-propellant estimate is essentially zero. That is the maximum payload value feasible for the mission case.**

*Thus it is payload that is determined in this analysis.***Orbits and the Associated Delta Vees**

As indicated in ref. 2,
I have looked at a Hohmann min energy transfer orbit, and 3 faster transfers with shorter flight
times. All of these are transfer
ellipses with their perihelions located at Earth’s orbit. For Hohmann transfer, the apohelion is at Mars’s orbit. For the faster transfers, apohelion is increasingly far beyond Mars’s
orbit. Why this is so is explained in
the reference. See Figures 3 and 4.

Note that the overall period of the transfer orbit is
important for abort purposes. If the
period is an exact integer multiple of one Earth year, then Earth will be at the orbit perihelion
point simultaneously with anything traveling along that entire transfer
orbit. This offers the possibility of
aborting the direct entry and descent at Mars,
if conditions happen to be bad when the encounter happens. Otherwise,
the spacecraft is committed to entry and descent, no matter what.

The cases examined in ref. 1 were all computed for Earth and
Mars at their average distances from the sun.
The larger transfer ellipse with the longer period occurs when both
Earth and Mars are at their farthest distances from the sun. This leads to larger delta vees to reach
transfer perihelion velocity for the trip to Mars, and larger velocity on the transfer orbit for
the trip back to Earth.

Ref 1 has the required velocities and delta-vees, but the most pertinent data are repeated
here:

__Transfer__

__E.depdV, km/s__

__trip time, days__

__M. Vint, km/s__

Hohmann 3.659 259 5.69

2-yr abort 4.347 128 7.40

No abort 4.859 110 7.36

3-yr abort 5.223 102 6.53

__Transfer__

__M.depdV, km/s__

__trip time, days__

__E.Vint, km/s__

Hohmann 5.800 259 11.57

2-yr abort 7.548 128 12.26

No abort 7.509 110 12.77

3-yr abort 6.653 102 13.14

I did not examine the worst cases for all the transfer
orbits in ref. 1, but I do have the increase in perihelion velocity for the worst
case Earth departure on a Hohmann transfer for Mars: 0.20 km/s higher than average. I also have the increase in apohelion
velocity for the worst case Mars departure on a Hohmann transfer for
Earth: 0.16 km/s higher than average.

I cheated here: I
used those worst-case Hohmann increases for all the faster trajectories as
well. That’s not “right”, but it should be close enough to see the
relative size of the effect of worst case over average conditions. I also used the same additive changes on the
entry velocities.

Because of the precision trajectory requirements for direct
entry while moving above planetary escape speed, some sort of

**. With this kind of analysis, I have no way to evaluate that need***course correction burn or burns will simply be required**. So I just guessed: 0.5 km/s delta-vee capability in terms of propellant reserves.*
Because this is just a guess, I did not run any sensitivity analysis on
it. However, the delta-vee budget proposed here is factor
2.5 larger than the difference average-to-worst-case for the trip to Mars, which suggests it is “plenty”. It is about factor 3 larger than the
difference average-to-worst-case for the return trip to Earth. You can get a qualitative sense of this
effect from examining that average-vs-worst case effect.

**Propellant Budgets for Direct Landings**

With this vehicle (or just about any other vehicle), entry

__must__be made at a shallow angle relative to local horizontal. Down lift is required to avoid bouncing off the atmosphere, since entry interface speed V_{int}exceeds planetary escape speed. This is true at both Mars and at Earth. Once speed has dropped to about orbit speed, the vehicle must roll to up lift, to keep the trajectory from too-quickly steepening downward.
The hypersonics end at roughly local Mach 3 speeds, which is around 0.7-1 km/s velocity, near 5 km altitude on Mars, and near 45 km altitude on Earth which has about the same air pressure. Up to that point, entry at Mars and Earth look very much alike, excepting the altitude. After that point they diverge sharply, as illustrated in Figure 5.

The descent and landing at Earth require the ship to
decelerate to transonic speed, then pull
up to a 90-degree angle of attack (AOA, measured
relative to the wind vector). Thus, as the trajectory rapidly steepens to
vertical, the ship executes a broadside
“belly-flop” rather like a skydiver.

At low altitude where the air is much denser, the terminal speed in the “belly-flop” will
be well subsonic. I assumed 0.5
Mach, but that might be a little conservative. This is the point where AOA increases to 180
degrees (tail-first), and the landing
engines get ignited. From there, touchdown is retropropulsive.

The landing on Mars is quite different. The ship comes out of hypersonics very close
to the surface, still at high AOA and
still very supersonic. From there, the ship must pitch to higher AOA and pull
up, actually ascending back toward 5 km
altitude. This ascent is energy
management: speed drops rapidly as
altitude increases. It’s not quite a
“tail slide” maneuver, but it is similar
to one.

At the local peak altitude,
the ship is moving at about local Mach 1, and pitches to tail first attitude, igniting the landing engines. From there,
touchdown is retropropulsive. The
Martian “air” at the surface is very thin indeed, as the figure indicates. It may be that thrust is required to assist
lift toward bending the trajectory upward:
the engines would have to be ignited earlier, and at higher speed, as indicated in the figure. Whether this is necessary is

__just not yet known__.
The low point preceding the local pull-up is at some
supersonic speed; I just assumed about
local Mach 1.5, as indicated in the
figure. That would correspond to a factor
1.5 larger landing delta-vee requirement,
implying a larger landing propellant budget.

In either case, I also
use an “eyeball” factor of 1.5 upon the kinematic landing delta-vee, to cover gravity loss effects, maneuver requirements, and any hover or near-hover to divert
laterally to avoid obstacles.

So, for purposes of
this sensitivity analysis, the Earth
landing is not of much interest, but the
Mars landing is.

*The sensitivity analysis looks at the effects of Mach 1.5-sized vs Mach 1-sized touchdown delta-vee.***Analysis Results**

The scope of the sensitivity analysis is illustrated in Figure
6. As indicated earlier, the orbital delta-vee

__increases__worst-case-vs-average, for Hohmann transfer, were applied additively to the departure delta-vees for the faster trajectories. No attempt was made to vary the course correction budgets. Growth in vehicle design inert mass was examined. An increase in the Mars touchdown delta-vee was examined. Nothing else was considered.**These results are given in Figure 7. These are the plots from the spreadsheet, copied and pasted into the figure. There are 4 such plots in the figure: the top two are for the outbound journey Earth to Mars. The bottom two are for the return journey Mars to Earth. Results for all 4 transfer orbit cases are shown simultaneously by using trip time as the abscissa.**

*The results start with the worst vs average orbital delta-vee sensitivity.*
Each has 4 data points:
these are for the Hohmann transfer at 259 days flight time, the 2-year abort orbit at 128 days, the non-abort orbit at 110 days, and the 3-year abort at 102 days. Be aware that the curves are probably not
really straight between the Hohmann orbit and the 2-year abort orbit.

*I did not run enough fast transfer cases in ref. 1 to get a smooth curve here.*
The most significant thing in the left hand figure for the
outbound trip is the about-40 ton loss of max payload between average and worst
case for the Hohmann transfer. This is a
lot less than the about-130 ton payload loss using the 2-year abort orbit
instead of Hohmann transfer, or the about-210
ton payload loss for using the 3-year abort orbit.

The average-vs-worst-case deficits are somewhat similar on
the faster orbits. The Mars entry
interface velocity trend in the right-hand figure is obviously very
nonlinear. Yet, all the calculated values fall below the
entry velocity from low Earth orbit (LEO).
Any heat shield capable of serving for return from LEO will serve this
Mars entry purpose, which would be the
governing case if the trip were one-way only.
There’s only a small change in entry speeds for average-vs-worst orbit case
in this estimated analysis.

The return voyage has trends shaped quite differently. For Hohmann transfer, the worst-vs-average payload loss is about 20
tons. The deficits on the faster orbits
should be similar. The deficit for using
the 2-year abort orbit instead of Hohmann is far larger at about 110 tons, and that’s from a small return payload to
begin with.

In the right hand Earth entry interface speed plot, the blue and orange curves in the entry
interface plot fall only slightly apart.
Note that all the entry velocities are much higher than the
just-below-escape speed seen with Apollo returning from the moon. The faster transfer orbits, and even the Hohmann transfer, are substantially more demanding than a lunar
return entry.

*It is clearly the direct-entry Earth return that will size the heat shield design!*

Figure 7 – Sensitivity to Worst-Case Orbital Distances vs
Averages

**This is the same 4-plot format as Figure 7. For the outbound trip to Mars, the Hohmann mass penalty for inert mass growth is about the same 40 ton deficit as for worst-case orbit distances. It is similar for the faster trajectories. It is the return trip that most suffers from vehicle inert mass growth. We lose about 40 tons from an already small return payload on the Hohmann transfer.**

*Results for the effects of inert mass growth sensitivity are given in Figure 8.***Note that both the Mars and Earth entry interface velocities are unaffected by this sensitivity. The orange and blue curves fall right on top of each other.**

*However the 2-year abort trajectory and the no-abort trajectory are entirely infeasible, with their max payloads calculated as negative.*Figure 8 – Sensitivity to Vehicle Inert Mass Growth

**This follows the same format as Figures 7 and 8. Bear in mind that the nominal design lights the engines for touchdown at about Mach 1 speed. For this analysis, the engines are ignited earlier, at about Mach 1.5 flight speed, to assist lift in pulling up to the Mach 1 “flip”, to tail-first attitude. That makes the landing delta vee about 1.5 times larger. (Note that each case is**

*The sensitivities to the need for a thrusted pull-up on Mars are given in Figure 9.*__also__factored up by 1.5 further, to cover any maneuver / hover needs for the touchdown.)

What the figure shows is about the same 40-ton payload loss
on the voyage to Mars to cover the increased landing propellant requirement for
the Hohmann transfer. Effects on the
faster transfers are similar. This trend
is comparable to the worst-case orbit losses.
The return payload is entirely unaffected, as the landing occurs prior to refueling and
loading for the trip home.

Both the Earth and Mars entry interface velocities are
unaffected by this Mars thrusted pull-up scenario. The orange and blue curves fall right on top
of each other.

**Final Remarks**

#1.

**Real 3-body orbital analysis, and real entry-trajectory lifting flight dynamics models, must be used to get better answers. Nevertheless, the trends are quite clear from this approximate analysis.***These results are only approximate!*
#2.

*Flying on faster transfer orbits will cost a lot of payload capability, on both the outbound voyage, and the return voyage.*__This effect is much worse on the return voyage__, where the allowable payload is just inherently smaller.
#3. The effects of
worst-case orbital positions-relative-to-average, of Mars and Earth, have a significant effect on payload, but it is only half or less the effect of
choosing faster transfer orbits.

#4. The effect of
vehicle inert mass growth from the design target of 120 metric tons to an
arbitrary but realistic 160 metric tons is comparable to the effect of
worst-case vs average orbits on the outbound voyage.

**This is enough to prevent faster-than-Hohmann transfers on the voyage home, for this vehicle model.***However it has catastrophic effects on the return voyage!*
#5. The effects of
needing a thrusted pull-up for the Mars landing is comparable to the effects of
worst-case orbit distances on the outbound voyage. This has no effects upon the return voyage.

#6.

**This is substantially more challenging than was the return from the moon. For deliberately-designed one-way vehicles to Mars, the heat shield design requirements are comparable to entry from low Earth orbit.***It is the direct Earth entry velocity that will design the vehicle heat shield for any vehicle capable of making the return.*
#7. My personal
opinions are that thrusted pull-up will be needed, along with the need to fly when Earth and
Mars orbital distances are worst-case,
plus there will be a little inert mass growth (say by 20 metric tons to
140 metric tons vehicle inert mass).

**Estimated performance data for this design case (at 140 metric ton inert mass) are in Figure 10 (same basic format as Figures 7, 8, and 9). Note that two of the faster transfers home are precluded. The feasible one has a very small max payload value compared to Hohmann transfer.***That kind of thing is the proper design point for this vehicle, not the most rosy projections!*
Figure 10 – Performance for Worst Orbits, Thrusted Pull-Up, and Some Inert Mass Growth

#8. Bear in mind that
the

**. The payloads for the faster transfers to Mars look more like what can be ferried up to LEO. That suggests that a faster transfer to Mars is most compatible with the projected “Starship” / “Super Heavy” system design characteristics, as these were evaluated in references 2 and 3.***rather high max allowable payload figures feasible to Mars for Hohmann transfer are incompatible with what can be aboard “Starship” for launch to low Earth orbit*
#9.

__Bear also in mind__that a**, in case conditions at arrival prove too bad to attempt the landing.***faster transfer orbit to Mars ought to include abort capability*__There is simply not the propellant available to enter orbit and wait for better conditions__. Thus life support supplies must be carried to last the__entire period of the transfer orbit__, and a full-capability heat shield for direct Earth entry__must__be used.
#10. The fast
transfer home need

__not__be limited by abort capability. It can be a different transfer orbit than the outbound trip. Surprisingly, the shapes of the plotted curves suggest that something faster than the “3-year abort” orbit could be used for the return home.
#11. Given a way to
combine two payloads to LEO into one “Starship” by cargo transfer operations on
orbit, then (and only then) the very
large payloads to Mars indicated for Hohmann transfer become feasible. Like on-orbit cryogenic refueling, this on-orbit cargo transfer capability does
not yet exist, not even as a concept
(on-orbit refueling at least exists as a concept).

**References:**

#1. G. W. Johnson,
“Interplanetary Trajectories and Requirements”, posted 21 November 2019, this site.

#2. G. W. Johnson,
“Reverse-Engineering the 2019 Version of the Spacex “Starship”/ ”Super
Heavy” Design”, posted 22 October
2019, this site.

#3. G. W. Johnson,
“Reverse-Engineered “Raptor” Engine Performance”, posted 26 September 2019, this site.

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