“Back of the Envelope” Entry Model

A note:  blogger took this down 4-16-2026 for an apparent complaint about "violating community standards".  This article had been published for nearly 14 years with no complaints!  The only change from what was published earlier is the 4-15-2026 update just below.  I looked through those standards and found not one standard that this technical article or that latest update violates,  not even copyright!  I gave attributions for everything.  I must conclude that any complaint was spurious.

-----   

Update 4-15-2026:  Note that this article was published long before Artemis-1 and -2 ever flew!  Yet it remains as accurate and useful as ever.  The stagnation heating models I use in the entry spreadsheet now include a plasma radiation model.  The original H. Julian Allen convective model I still use.

Q/A_conv = 1.75E-8 * [(rho, kg/m³)/(Rn, m)]^0.5 * (V, m/s)³

Q/Arad = 27.94 * (rho/rho0)^1.7 * [(V, km/s) / 3.048]^12.5 

(where rho0 refers to scale height model for density built into the atmosphere models)

I typically just scale down the stagnation values for rough estimates of heating rates away from stagnation.  For still-attached flow,  I use stagnation convective/3 and just use the stagnation radiative as it is because the plasma sheath and bow shock are still very close to the surface.  For separated wake zones,  I use stagnation convective/10,  and stagnation radiative/3 because the plasma sheath is more remote from the surface,  as is any shock.  Those are just ballpark guesses on my part.

The accuracy of that spreadsheet entry model (and my Apollo ballistic coefficient and entry angle data) is attested-to by the actual entry time of ~6 minutes from entry interface to end-of-hypersonics at Mach 3,  the end-of-entry altitude (near 120-125,000 feet),  and the peak deceleration gees (Apollo was 10-11 gees).  It does not do skip entries,  just straight-in.

This is now the way I take the plots the spreadsheet creates,  and display them as a representation of what happens during the hypersonics:

-----  

Original article and earlier updates follow:

-----   

In a previous article, realistic atmosphere models were determined, and extended to include speed-of-sound profiles (see “Atmosphere Models for Earth, Mars, and Titan”, dated 6-30-12). These were based on atmosphere data reported by Justus and Braun (reference 1). Those Justus and Braun atmosphere models included recommendations for a density scale height model, also reported in the 6-30-12 article. This would be used for simplified entry dynamics and heating calculations, the topic here.

Update 1-21-13:  Please see the newer posting,  dated 1-21-13 and titled BOE Entry Model User's Guide,  for more details on how to set up and use this spreadsheet model.  

Update 3-23-13:  I have since found that the simplified entry model described in the Justus and Braun paper actually refers to the early 1950's work of H. Julian Allen at NACA.  He was able to publish this model openly in the mid-1960's,  after it was declassified.

Density-Scale Height Model

As reported in reference 1, for regions where the density scale height is relatively constant, there is a simple exponential correlation of density versus altitude. The density scale height is defined as density divided by its own gradient (with altitude):

Hρ = ρ / (dρ/dz) where z is altitude

In regions where this scale height is at least approximately constant, the relationship between altitude and density is well-approximated by a two-parameter fit:

ρ(z) = ρ(0)*exp(-z/Hρ) where “exp” is the base e exponential function

In this equation, ρ(0) is a curve fit parameter only, and may not correspond to surface density at all. The values of ρ(0) and Hρ applicable to entry altitudes for Earth, Mars, and Titan were reported previously in the 6-30-12 article, exactly as given in reference 1.

Vehicle Model

In hypersonic flow, the drag coefficient of an object is essentially a constant across a broad range of flight Mach numbers, from entry speeds down to a “low limit”. That low limit is roughly Mach 3 for a blunt object, and about Mach 5 for “sharp” or “pointy” objects. Most capsules entering blunt-heat-shield-surface-first are “blunt” objects, with a lower Mach limit of about 3, for applicability of any and all hypersonic flow models.

In most models, the parameter of interest is the ballistic coefficient β, which is computed from the vehicle mass m, blockage area A, and hypersonic drag coefficient CD, as below. The usual units these days are kg for mass and square meters for blockage area. CD is nondimensional.

β = m / CD*A

The values of β tend to be smaller numbers at smaller masses, and larger at higher masses. That scaling with vehicle mass is a topic for another article. The largest value in reference 1’s study was β = 200 kg/sq.m, and that is the value used here. It is larger than that for all probes sent to Mars so far.

The larger the β, the deeper into the atmosphere the body penetrates during the hypersonics. This also acts to raise maximum gees and the heating experienced during the entry.

Entry Model

There are three important parameters, all associated with the location of the “atmospheric interface”, where deceleration effects due to drag begin. This is a “fuzzy” notion, so the choice is a bit arbitrary. It is usually based on a density value. The geometric altitude at interface zatm, varies from one celestial body to another. The velocity at interface Vatm can vary widely, from above escape Vesc to below circular orbit velocity, depending upon circumstances. Vesc itself should be adjusted from the surface value to the interface altitude:

Vesc at zatm = surface Vesc * (R / (R + zatm))^0.5 where R is the body’s radius

The circular orbit velocity is very simply related to the escape velocity. For entry analyses, this should be done at zatm. After a de-orbit burn, the actual entry velocity will be very little different from the circular orbit velocity.

Vcirc at zatm = Vesc at zatm / (2)^0.5

For vehicles coming in from deep space for a faster-than-escape entry, there is a velocity V∞ typical of its interplanetary orbit, in the absence of the influence of the local body’s gravity. As the vehicle approaches the body, its gravity affects vehicle relative velocity at interface. This effect can be computed from kinetic energies:

Vatm = (V∞^2 + Vesc^2)^0.5 where Vesc is the value at zatm

There is also an entry angle θ from local horizontal. This is best visualized in “flat Earth” Cartesian coordinates. The simplified entry model presumes the trajectory to be a straight line for non-lifting vehicles, and that really is a realistic notion for the hypersonics. From the change in altitude and the tangent of θ, the horizontal change in range is calculated. Using instead the sine, one may compute the change in slant range down the trajectory line. These increments can be summed up. See also figure 1 below. The “end” of this analysis would be when velocity drops to about local Mach 3, computed as velocity divided by sound speed at that altitude. (That does not have to be very precise.)

The steeper the θ (angle below horizontal), the deeper into the atmosphere the vehicle penetrates during entry. This acts to raise deceleration gees and heating, plus the altitude at end-of-hypersonics is lower.

Here are the entry interface altitudes recommended in reference 1:

zatm, km body
140 Earth
135 Mars
800 Titan

The entry velocities I used in my study here are just escape speeds at interface altitude, typical of a V∞ = 0 situation for Earth and Mars. The escape speed on Titan is not even hypersonic, so I chose an utterly arbitrary value, just to have some numbers to analyze. The surface escape speeds are also given, so you can see the effect of scaling to zatm for Earth and Mars. These values are:

Vatm, km/sec body surface Vesc, km/sec
11.058 Earth 11.179
5.026 Mars 5.0282
3 (arbitrary) Titan 0.8161   UPDATE 2-20-17 should be 2.58 km/s

Velocity Trend from Simplified Model


Velocity down the trajectory is related to the scale height and vehicle parameters by:

V(z) = Vatm * exp(-C * exp(-z/Hρ)) where z and Hρ are km, and the velocities are km/sec
C = (ρ(0) * Hρ * 1000)/(2*β*sinθ) where Hρ is km, ρ(0) is kg/cu.m, and β is kg/sq.m

In my version, θ is positive downward, requiring the negative sign on C in the argument of the exponential in the V equation. This is different from the positive upward convention in reference 1, where the sign on C was positive, because its value was already negative, due to the sign on the sine. Reference 1 was “hazy” on units of measure, but those are clarified here. The “1000” in the C equation given here converts scale height units from km to meters, so that C may be dimensionless. Reference 1 does not show that.

Repetitive calculations across a spread of altitudes (from zatm downward) produce a list of V values, one for each z. The corresponding horizontal and slant ranges may be computed from zatm and θ as below. This leads to a list of velocities versus slant ranges that may be plotted. This is the velocity profile down the slant line trajectory, something not presented in reference 1, but actually quite useful.

R(z), km = (zatm – z, km)/tanθ
S(z), km = (zatm – z, km)/sinθ

From one point to the next going downward through the list vs altitude, velocities may be averaged, and the slant range distances may be differenced. Slant range distance increment divided by average velocity in that increment is the time increment from the one point to the next in the list. These may be summed from 0 at interface to produce a timeline down the trajectory. If the list of altitudes is fine enough, these timeline calculations can be rather accurate. Because altitudes are km and velocities are km/sec, times in this timeline will be seconds. Entry can be hundreds to thousands of seconds long (a few to several minutes).

A profile of deceleration gees may be computed using that timeline. From one point to the next, the difference in velocities divided by the difference in times is the acceleration in absolute units. Times will be seconds, but velocities must be in m/sec, requiring 1000*km/sec values. If you do this, accelerations will be in m/sec2. Divided by the standard value 9.8066 m/sec2, this is deceleration gees. This produces a list of gees vs. slant range as a deceleration profile down the trajectory. It invariably shows a peaked behavior. The peak value value computed this way compares very closely with the closed-form max gee equation used in Reference 1. The old model also includes closed-form equations for the velocity and altitude at which gmax occurs. These correlate well with my numerically-accumulated solution, leading to the conclusion that Justus and Braun modeled the deceleration dynamics rather well in their paper.

Convective Heating Model

There actually are some useful simple models for stagnation point convective heating during entry. Radiation heating is not so simple, but does range from a significant to a dominant effect. The incandescent dissociated gas adjacent to the spacecraft is, quite simply, a very bright, hot fire warming the surface. In the absence of any calculations at all, a rough-and-ready rule of thumb might be to triple the convective values. However, that is not a proper basis for actual design. Neither are convective values alone. “Fixing” this is beyond scope in this article.

Reference 1 used one of the simplest of the old convection correlations. It relates stagnation point heat flux to nose radius rn, ambient density ρ, and velocity V. This is a dimensionally-inconsistent correlation equation with a constant of proportionality k that includes the conversion of the units of measure. I found very serious problems with the use of this equation in the heating analysis of reference 1. One problem was the identification of this constant of proportionality k as a “heat transfer coefficient”, which it most definitely is not. Here is the correlation equation:

q = k (ρ/rn)^0.5 V^3

I found in a much older reference (2) this same correlation set up for American units of measure. Nose radius was feet, velocity feet/sec, density lbm/cu.ft, and heat flux BTU/sq.ft-sec. For these units, k was numerically 3.16 x 10-9. Carefully converted to metric units (nose radius m, velocity m/sec, density kg/cu.m, and heat flux W/sq.cm), k should numerically be 1.748 x 10-8. That is not what I found back-calculated from the heating data reported in reference 1. They were using 1.006 x 10-8 for the peak heat flux model, and 1.575 x 10-15 for the total heat absorbed model. Those differ by 7 orders of magnitude, not the 3 orders of magnitude that account for the change from W-sec = J to KJ units.

The old analysis reported in reference 1 did not compute heating down the trajectory, only the closed-form equations for peak heating and the velocity and altitude at which it occurs, plus a closed-form equation for total heat absorbed during entry (representing the time integral of the heating rate). The equations presented for this in reference 1 show the same variables k, rn, plus ρ(0), Hρ, and Vatm. It’s supposed to be the same k, but it is not the same k in reference 1.

For the units they used, the two k-values should have differed only by the factor of 1000 to convert J to KJ units. Both were per sq.cm, and per sec of time in the flux. Reference 1’s actual k numbers differed in the significant digits between flux and total heat, plus 4 extra orders of magnitude in the power of 10. The one used for heat flux differed from the correct value by about a factor of 1.5, which is not all that significant, although their error is in the under-prediction (wrong) direction. They also used different values at Mars from those at Earth. These numbers strongly resemble arbitrary selections to match a known heating point to the corresponding variable values in the equation, not consistent use of the correlation.

I did not do any of that. I used the correct metric k-value in the heat flux correlation, directly in my developed numerical trajectory list, producing a heat flux value at each point analyzed. My heating peak value differs from reference 1’s closed-form estimate by the ratio of the k-values used. My altitude and velocity at peak heating resemble those of reference 1. That part of the reference 1 heating estimates is good to the factor 1.5 difference in k-values that we used.

I numerically integrated my heat flux values with time, using the timeline developed in the list. My integrated total heat does not match the closed-form estimate at all (I did that, too). The significant digits are close, but my closed-form data show a 7-orders-of-magnitude problem with the powers of 10. I cannot find the error in the closed form equation for total heat, but it is there. The integrated total does look reliable, though, in my trajectory listing. I suspect that this discrepancy in the answers from the closed-form equations versus integrated totals is why no one used this old model for any heating estimates. Corrected like this by numerical integration, it now looks good to me. Do not use the closed form heating estimates, use the trajectory listing and integrated total instead. This is easy to do in a spreadsheet.

The Trajectory Selection Process

I did not iterate to optimize entry trajectory in any way for this article. For a true design study, one would follow the sequence in figure 2 below to determine the Mach 3 “end-of-hypersonics” slant range and velocity from the velocity vs. slant range plot. Those endpoint conditions can be transferred to the other charts, which determines “end-of-hypersonics” altitude, from the altitude vs. range plot. The peak gees can be read from the deceleration gees vs slant range chart. Peak convective heating rate and total convective heat absorbed can be read from the heating vs. slant range chart. The iteration is on peak gees and altitude at end-of-hypersonics, which have practical limits (max gee, min altitude). The heating parameters just feed into a thermal protection system (TPS) design process. The variables you change to optimize your trajectory selection are θ, β, and perhaps the selection of Vatm.

Calculated Data for Unoptimized Trajectories

I ran escape-speed models for Earth and Mars, at 1 degree below horizontal and β = 200 kg/sq.m. These compare directly to the 1 degree, β = 200, V-at-infinity= 0 cases in reference 1. For Titan, my arbitrary Vatm was 3 km/sec, so those data do not compare to anything in reference 1.

The sequence of figures for Earth entry is figures 3, 4, 5, and 6 below. These are presented in analysis sequence order. Figure 3 is the velocity profile vs slant range plot. Using the speed-of-sound profile for Earth’s atmosphere, one can select which point in the tabulated list is closest to Mach 3. (Mach number is velocity divided by sound speed, same units for both.) Sound speeds in that part of the atmosphere are around 300 m/s, so 1 km/sec is pretty close to Mach 3. The spreadsheet list (imaged in figure 7) tells us which altitude is the end-of-hypersonics altitude. That point is then marked in the figure. We also have the end-of-hypersonics slant range from the list.

The second plot is range and slant range vs altitude. At only 1 degree depression, these two happen to fall just about on top of each other in the figure. The Mach 3 point can be marked using the altitude from the list, as selected from the previous figure.

The third plot is the deceleration gees vs slant range profile. End of hypersonics can be marked with the slant range determined from the Mach 3 point above. The peak deceleration gee can be located on the graph, and picked out of the list, including its corresponding altitude and velocity.

The fourth plot presents the heat flux and integrated total heat vs slant range profiles. End of hypersonics can be marked with the slant range determined from the Mach 3 point above. The peak convective stagnation point heat flux can be located on the graph, and picked out of the list, including its corresponding altitude and velocity. The integrated total at the end-of-hypersonics is the value at the end-of-hypersonics point (the Mach 3 point).

For Earth, I could enter steeper and still stay below the 11 gees we used in Apollo returning from the moon. This shouldn’t be a problem for too low an end-of-hypersonics altitude, as that was over 40 km for 1 degree. This study has not yet been run by me, but would produce different outcomes for each β and Vatm selection. Having a “map” of these variations quickly was the advantage of the way this type of analysis was done in reference 1. For the dynamics, reference 1 provides good data; just not on the heat transfer.

The same sequence of plots for Mars is given in figures 8, 9, 10, and 11 below. Results look similar, except that gees and heating are much lower. So also is altitude at end-of-hypersonics. For Mars, I could enter steeper at higher gee, but this would put the end-of-hypersonics altitude even lower, and it is already fairly low. Higher β just makes this quandary even worse. That is the basic problem with traditional entry and supersonic/subsonic deceleration on Mars: little or no solution space adequate for effective chute-assisted descent. That spreadsheet is imaged in figure 12.

Even lift during the entry hypersonics won’t help that picture very much. Reference 3 discusses some ways and means around this quandary. Personally, I favor using lift and retro thrust through entry, and thrust all the way to touchdown; probably low thrust during the entry, and low thrust with the chute (if any), then high thrust to touchdown. Thrust offsets some of the mass’s inertia during deceleration, effectively lowering ballistic coefficient. There are many practical problems with this, of course. Those are out of scope here.

Finally, the same sequence of plots for Titan is given in figures 13, 14, 15, and 16 below. In this case, gees and heating are very low and end-of-hypersonics altitude is hundreds of km above the surface, in a very dense and deep atmosphere. A first impression says that Titan entry could be fairly easy to achieve with very high-β vehicles, at very high direct-entry speeds, and steeply-diving trajectories. All of that needs to be explored. The Titan spreadsheet is imaged in figure 17.

Conclusions Regarding “Common Lander Designs”

A really interesting design goal here might be a “common lander” that could land on Mars, or Titan, or any of the airless low-gravity worlds. Any design that works on Mars should work on Titan, and also on the airless worlds if adequate retro thrust capability is included. (It is the one of those airless worlds with the highest escape velocity that will size that propellant requirement.) One might also design for descent only, or for descent-ascent, separately. Nuclear propulsion might make a single-stage reusable descent-ascent design possible. These are all topics for future analyses and articles.

References

1. “Atmospheric Environments for Entry, Descent, and Landing (EDL)”, C. G. Justus (NASA Marshall) and R. D. Braun (Georgia Tech), June, 2007.
2. “SAE Aerospace Applied Thermodynamics Manual”, SAE Committee AC-9, published 1969 by the Society of Automotive Engineers.
3. “Mars Exploration Entry, Descent, and Landing Challenges”, R. D. Braun and R. M. Manning, no date except latest cited reference 2006.



Figure 1 – Simplified Entry Trajectory


Figure 2 – The Trajectory Analysis and Optimization Process


Figure 3 – Earth Entry Velocity Profile vs. Slant Range


Figure 4 – Earth Entry Range and Slant Range vs. Altitude


Figure 5 – Earth Entry Deceleration Gee Profile vs. Slant Range


Figure 6 – Earth Entry Heating Traces vs. Slant Range


Figure 7 – Image of Earth Entry Spreadsheet Data


Figure 8 – Mars Entry Velocity Profile vs. Slant Range


Figure 9 – Mars Entry Range and Slant Range vs. Altitude


Figure 10 – Mars Entry Deceleration Gee Profile vs. Slant Range


Figure 11 – Mars Entry Heating Traces vs. Slant Range


Figure 12 – Image of Mars Entry Spreadsheet Data


Figure 13 – Titan Entry Velocity Profile vs. Slant Range


Figure 14 – Titan Entry Range and Slant Range vs. Altitude


Figure 15 – Titan Entry Deceleration Gee Profile vs. Slant Range


Figure 16 – Titan Entry Heating Traces vs. Slant Range


Figure 17 – Image of Titan Entry Spreadsheet Data

Atmosphere Models for Earth, Mars, and Titan

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. 

----------     

Update 6-5-2016:  this is one of the most popular articles on the entire website.  I hope this publicly-available information has proven useful to others.
-------------------------------------------------------

These data replace the previous post "Mars Atmosphere Model", dated 6-24-12, just below.

Atmosphere Models for Earth, Mars, and Titan GWJ 6-30-12

After perusing several links on the internet, I settled on a June 2007 paper by C. G. Justus and R. D. Braun (ref. 1), which contained rather realistic “typical” atmosphere data for Earth, Mars, Titan, Venus, and Saturn. For manned landings, Earth, Mars, and Titan are the only destinations of interest that have atmospheres. Justus is involved with the site-tailorable EarthGRAM and MarsGRAM models. On both Earth and Mars, site-specific atmosphere models can vary widely, which is what the –GRAM models calculate. For Titan, the Huygens descent probe data are really all we have.

The cited paper (ref. 1) has near-surface atmospheric composition, average molecular weight, and specific heat ratio data, for each world, reproduced here for the three destinations of interest. The report also gives an indication of atmospheric composition versus altitude for each world. For my purposes here, that devolves to an altitude limit above which the surface composition is no longer at all representative.

There are also recommendations for high-altitude average density criteria at which aero-braking and aero-capture apply. These are 4x10^-7 kg/m^3 and 2x10^-3 kg/m^3, respectively. Aero-braking refers to multi-pass atmospheric drag braking into a desired non-escape orbit, and aero-capture refers to one-pass atmospheric drag braking into a non-escape orbit.

For purposes of estimating when significant heating during entry begins, versus significant simple drag effects, the aero-capture altitude is the better measure. Simple drag effects “begin” closer to the aero-braking altitude cited. This is true for all three worlds covered by this document: Earth, Mars, and Titan. Only the relevant altitudes vary.

Earth

Earth’s atmosphere has been modeled in many different ways, all both site- and season-dependent. Some are well-known. These include the US 1962 standard atmosphere, the extended ICAO standard atmosphere, the US polar and cold atmospheres, and the US tropical and hot atmospheres. Some are not so well-known, such as the Max Ambient In-Flight and Min Ambient In-Flight days, given in older versions of the Pratt and Whitney vest-pocket aeronautical handbook. Only the ICAO model extends to 100-km altitudes. Continuum flow models are pretty much inapplicable above roughly 45 or 46 km, which corresponds to the aero-capture altitude.

The density profile can be approximated between two specific altitudes as a simple exponential model dens(z) = dens(0)*exp(-z/Hd), where z is altitude, dens(0) is a curve fit parameter, and Hd is the density scale height. Average density scale height between those two altitudes is estimated by Hd = (z2 – z1)/ln(dens1/dens2). This is applicable to computations of max-deceleration, max-dynamic pressure, max heat rate flux, and max absorbed integral heat flux, all as functions of ballistic coefficient and entry velocity. Those effects I will explore in another posting.

The version reported here is an abbreviated table from the cited paper, and 6 plots, given as figures 1 through 6 below (at end of this article). They include the temperature profile, the pressure profile, the density profile, the speed-of-sound profile, the scale height profile, and a representation of the abbreviated table of values. It resembles the extended ICAO atmosphere model.

There is also a short table just below representing near-surface composition, properties, and scale height models. I interpolate the aero-brake altitude as 92.66 km, versus the aero-capture altitude as 46.72 km. Pressure data in are missing in the reported table and figure above the altitude at which composition varies significantly from surface values.

Table 1 – Earth Atmospheric Surface Parameters

for surface air composition 78.084% N2, 20.946% O2, Ar 9340 ppm, CO2 350 ppm, Ne 18.18 ppm, He 5.24 ppm, CH4 1.7 ppm, Kr 1.14 ppm, H2 0.55 ppm
usually near 1% water vapor

MW Cp/Cv
28.97 1.4

best fit 0-100 km
d(0)kg/m^3 = 1.226
Hd km = 7.256


Mars

The tables and figures are “representative” of Mars, knowing full well that variations there are more severe than on Earth. The data quoted herein are within about half an order of magnitude correct, versus a few percent for Earth. These data are given in figures 6-12 below, plus the following abbreviated table. I interpolate the aero-braking altitude as 94.55 km (comparable to Earth), and the aero-capture altitude as 23.82 km (much lower than on Earth). Pressure data are missing in the reported table and figure above the altitude at which composition varies significantly from surface values.

Table 2 – Mars Atmospheric Surface Parameters

for surface "air" composition CO2 95.32%, N2 2.7% , Ar 1.6%, O2 0.13%, CO 0.08%, H2O 210 ppm, NO 100 ppm, Ne 2.5 ppm, HDO 0.85 ppm, Kr 0.3 ppm, Xe 0.08 ppm

MW Cp/Cv
43.34 1.33

best fit 25-70 km
d(0)kg/m^3 = 0.03032
Hd km = 8.757


Titan

The tables and figures are based on Huygens descent probe data. They are the best we have. These data are given in figures 13-18 below, plus the following abbreviated table. I interpolate the aero-braking altitude as 637 km, and the aero-capture altitude as 192 km. Titan is unusual in having a far deeper atmosphere, of higher surface pressure and a lot higher density (because it is so cold there), than on Earth.


Table 3 – Titan Atmospheric Surface Parameters

for surface "air" composition N2 97.7%, CH4 2.3%

MW Cp/Cv
27.32 1.4

best fit 130-800 km
d(0)kg/m^3 = 0.1006
Hd km = 48.38


References:

1. “Atmospheric Environments for Entry, Descent, and Landing (EDL)”, C. G. Justus (NASA Marshall) and R. D. Braun (Georgia Tech), June, 2007.
2. “Atmospheric Models for Mars Aerocapture”, C. G. Justus, Aleta Duvall, V. W. Keller, no date except latest cited reference 2005.
3. “Mars Global Atmospheric Reference Model (MarsGRAM 2005) Applications for Mars Science Laboratory Mission Site Selection Processes”, H. L. Justh and C. G. Justus, no date except latest cited reference 2005.





Figure 1 – Abbreviated Earth Atmosphere Table


Figure 2 – Earth Atmosphere Temperature Profile


Figure 3 – Earth Atmosphere Pressure Profile


Figure 4 – Earth Atmosphere Density Profile


Figure 5 – Earth Atmosphere Speed of Sound Profile


Figure 6 – Earth Atmosphere Density Scale Height Profile


Figure 7 – Abbreviated Mars Atmosphere Table


Figure 8 – Mars Atmosphere Temperature Profile


Figure 9 – Mars Atmosphere Pressure Profile


Figure 10 – Mars Atmosphere Density Profile


Figure 11 – Mars Atmosphere Speed of Sound Profile


Figure 12 – Mars Atmosphere Density Scale Height Profile


Figure 13 – Abbreviated Titan Atmosphere Table


Figure 14 – Titan Atmosphere Temperature Profile


Figure 15 – Titan Atmosphere Pressure Profile


Figure 16 – Titan Atmosphere Density Profile


Figure 17 – Titan Atmosphere Speed of Sound Profile


Figure 18 – Titan Atmosphere Density Scale Height Profile

Mars Atmosphere Model

The NASA Glenn Research Center (Glenn RC) has published online a model of the properties of the Martian atmosphere.  It was based on measurements made by the Mars Global Surveyor in 1996.  The basic model comprises a temperature profile with two lapse rates (one below 7 km and the other above 7 km),  and a single exponential fit of pressure versus altitude.  Density is computed from these profiles. 

These Glenn RC data are available in either metric or US customary units at:  

http://www.grc.nasa.gov/WWW/k-12/airplane/atmosmrm.html   (metric)

http://www.grc.nasa.gov/WWW/k-12/airplane/atmosmre.html  (US customary)

One problem with this model is extending the upper zone lapse rate to very high altitudes.  Above about 115 km,  this model predicts temperatures below absolute zero (0 degrees K,  -273.15 degrees C).   I “corrected” that by limiting the extreme altitude temperatures to a constant 4 degrees K (-269.1 degrees C) as indicated in Figure 1.  But,  I do not believe this profile is anywhere near correct at altitudes around 100 km,  because the temperature profile in the extended ICAO atmosphere model for Earth does not tend to very cold temperatures at very high altitudes.  It actually goes very hot.

UPDATE 6-30-12:  there are far better models to use,  such as MarsGRAM.  Do not use this one!  There is a published paper "Atmospheric Environments for Entry,  Descent,  and Landing (EDL)" by Justus and Braun that has better temperature profiles for Mars and other destinations.  It also has a scale height-based way of approximating density,  and some 1956-vintage back-of-the-envelope entry hypersonics relations.  These are likely within half an order of magnitude of the correct design conditions.  In future postings,  I will elaborate a better "typical model" for the atmosphere of Mars,  and what kinds of vehicles might land large payloads through it.     

SECOND UPDATE 9-25-12:  The Justus and Braun atmosphere models (more than just Mars) and 1956-vintage entry hypersonics model,  I have examined,  updated,  and posted as usable "how-to" articles posted just above.  For atmospheres,  see "Atmosphere Models for Earth,  Mars,  and Titan" dated  6-30-12.  For the entry hypersonics models,  see "'Back of the Envelope' Entry Model",  dated 7-14-12.  Those two have what you need,  in all likelihood. 

Nevertheless,  using the “corrected” Glenn RC temperature profile and the Glenn RC pressure profile (Figure 2) to a peak altitude of 200 km,  I computed a density profile  vs altitude (Figure 3),  and a speed-of-sound profile (Figure 4).   This speed of sound was based on pure CO₂ at a molecular weight of 44.01,  and a specific heat ratio (γ) of 1.300.  It shows ridiculously-low values around 100 km,  precisely because the predicted temperatures are ridiculously low. 

It is known from experience on Earth that “re-entry begins” around 90 km altitude.   It is a little difficult to quantify why that might be so.  The density in the ICAO atmosphere at 90 km is about 3.213 x 10^-6 kg/m³.   The corresponding mean free path length between air molecules is about 3 cm.  Above 90 km,  the molecular weight departs from 28.966,  reflecting a change in composition.   Aerodynamically,  this is most definitely free molecule flow,  probably best represented by some modified Newtonian flow model.  Accordingly,  I calculated a Newtonian stagnation pressure (P_stag) of 2 mbar at 90 km,  and 7905 m/s velocity,  which is Earth’s surface circular orbit velocity.  Actual entry speeds from real orbits are not much different from that value at all. 

In the Mars atmosphere model presented here,  I computed an Earth / 90 km-equivalent density of 3.213x10^-6 kg/m³ at about 140 km on Mars.   I computed an equivalent P_stag = 2mbar for Mars at 122.2 km,  using a Mars surface circular orbit velocity of 3636 m/s,  and a profile I generated (Figure 5). 

I personally have no way to compute mean free path on Mars.   So,  based on either density,  or Newtonian P_stag,  re-entry should start a little higher up on Mars:  somewhere near 120-140 km,  versus about 90 km on Earth.  This value is comparable,  so there is some confidence in this assertion. 

Now,  in Earth’s atmosphere,  one starts to lose the applicability of the continuum flow model somewhere around 150,000 feet (45-46 km).  That density is around 1.48x10^-3 kg/m³,  and the corresponding mean free path is around 0.1 mm.  These data are available in many references;  the one I used was the CRC Handbook of Chemistry and Physics,  53^rd Edition,  1972-1973,  pages F171-F174.

In the Glenn RC model for Mars,  that same density occurs at roughly 28 km altitude.   Accordingly,  I would use some sort of Newtonian flow model for altitudes above 28 km,  and the continuum flow model below that altitude.  This choice is also speed-dependent,  however.   In any event,  I am not sure this is a model adequate for predicting hypersonic drag during high-altitude re-entry.  The temperatures “look too low” up there.   The computed Mach numbers will be all wrong. 

On the other hand,  for the final supersonics and chute-deployment phases,  somewhere below roughly 28 km altitudes,  this model is probably pretty good. 

Figure 1 – NASA Glenn RC Temperature Profile Model, Limited to 4 K Minimum

Figure 2—NASA Glenn RC Pressure Profile

Figure 3 – Density Profile Based on NASA Glenn RC Model

Figure 4 – Speed of Sound Profile Based on NASA Glenn RC Model

Figure 5 – Newtonian-Flow Stagnation Pressure Profile at Constant Circular Orbit Velocity

Quote from MSNBC news Monday 6-21-12:

"Taliban commander in Pakistan’s tribal belt says that the U.S.-funded vaccinations for tens of thousands of children would be outlawed until drone attacks stop."

Observation:

These bastards are now hiding behind diapers,  not just skirts.  They're really afraid of drone strikes.

Conclusion:

Strike more,  not less.  Lots and lots and lots more.  The world will be a much better place once they're dead.  Don't stop until they are. 

Corollary:

Most of the societies that tolerate groups like this in their midst will be afflicted with really nasty and evil violence,  until they learn better than to tolerate this nonsense. Until then,  they are not civilizable into fit company.  For anybody. 

Kudos to the Chinese

My congratulations to the Chinese for their launch of Shenzou-9 with 3 aboard,  including their first woman astronaut.  Their manned spaceflight record seems pretty good so far.  We'll soon see if the docking goes well,  along with their first cut at space station-like operations.  Best wishes and Godspeed.

GW

Pressurizable Domed Habitat Structures

The image is a diagram of a spherical-segment membrane-type pressure dome structure, such as might be used on Mars to pressurize acreage for agriculture in a future colony.

The membrane is a uniform tensile stress tension-field structure, tied to a retention ring buried below grade in the surface. The "blowout load" on ring diameter is balanced by the weight of the ring structure, which is in compression in bulk, but locally tension where the membrane ties to it. The tension stress in the membrane is proportional to the spherical radius, and to the net pressure differential across the membrane.

The membrane itself ought not to be a single layer. The material needs to let some visible and some UV light through for photosynthesis, but must be proof against UV damage over time. (This pretty much rules out all known polymers.) Access to that layer actually holding the gas pressure must be unimpeded from below, in order to repair meteroid punctures, which will occur. That's a basic safety thing.

The retention ring would be reinforced concrete if it were built on Earth. Concrete as we know it won't set in the cold on Mars, so I suggest "icecrete". That's a composite of water ice as the matrix, sand,  and rounded-by-tumbling rocks as the aggregate, and steel (or other comparably stiff) reinforcing bar. You mix the aggregate up with liquid water, pour into forms containing the reinforcement bars, and let it freeze. It either must be coated or buried to prevent sublimation of the ice.

One should note that the weight per unit circumference of the retention ring scales with diameter of the ring (and its cross section area), whereas the "blowout" load scales as ring diameter squared. For any given collection of materials, there will be a maximum size that can be practically built. This is because the increases in the retention ring cross section area cannot make up the discrepancy between the ring diameter and ring diameter squared factors.

One should also note that the shear stirrups in the ring "beam" are also the tensile connections between the membrane and the circumferential reinforcing bars inside the ring. Those stirrups ought to "loop" at least some of those circumferential bars, as shown in the image. I do not know the details of connecting these stirrups to the membrane, but I do think there ought to be a lot of these stirrups (i.e., the spacing is very close).

I do not have any actual design numbers available. But these are the fundamental structural and safety design criteria.

Have fun .....

GW

UPDATE 1-26-13:  See also "Aboveground Mars Houses" dated 1-26-13 for a pressurized building concept that is much closer to an in-hand,  practical technology.  

-----   

search code DDMMYYYY            09062012

search keywords                              space program, Mars

-----   

Deceleration by Drag Devices (and More) on Mars

The “classic” landing scenario on Mars dates all the way back to Viking in 1976. After hypersonic entry is done, the vehicles “pops out” at around Mach 2.5, ready for ribbon chute deployment of one type or another. On Earth, that’s up around 40,000 feet (13 km); on Mars, it’s substantially closer to the surface. That’s because, even with the lower gravity, the thin “air” of Mars is far less effective as a hypersonic entry decelerator.

This hypersonic decelerator effect is more-or-less proportional to density, and to speed squared during entry. Entry speeds on Mars are less than those on Earth, due to the lower gravity. But, “air” densities are very, very far lower. The density effect “outweighs” the gravity effect by quite a lot.

Once the chutes are out, it becomes a simple drag-to-weight ratio question, all other things being equal. On Earth, densities are far higher, but so is weight. The surface density on Mars varies with temperature, but is around 0.6% of that here. These densities reduce essentially with atmospheric scale height, but the overall density argument still applies.

The surface gravity on Mars is 38% of that here. Drag/weight for a chute then scales as density/gravity, for 0.6%/38% = about 2%. So chutes are simply far less effective as decelerators on Mars, as measured in the same velocity range, from about Mach 2.5 to impact.

Drag/weight for a parachute system varies directly proportional to chute blockage area, drag coefficient, air density, and velocity squared, and inversely with load weight. Kinematically, we’d like the terminal velocity of a chute-and-load system on Mars to be similar to that here, around 20-30 mph (30-50 kph). That leaves only density and weight, which ratio at about 2% for mars. See Figure 1 below.

In other words, for the same kinematic effect, the payload retarded by a chute on Mars can be about 2% of what we would expect for the same size chute on Earth. (It can be more if you can accept a higher terminal velocity, except that on Mars, you cannot. You are already too close to the surface when the atmospheric entry hypersonics are over.)

For ribbon chutes, which work at moderate supersonic Mach numbers all the way down subsonic, the drag coefficient is in the range of 1 to 1.5 and is a function of Mach number, peaking near Mach 1.1-ish. It is not substantially different with ballutes, or any other inflated structures. To get the drag to support the load at a reasonable terminal velocity, the decelerator must be around 50 times larger on Mars than on Earth. This is one real whopper of a landing design problem!

The only way out of this dilemma that I see, is a combination of aero decelerator braking, and rocket retro thrust braking, done simultaneously. That is something we have not yet attempted on Mars.

But, it is something “we” (but not all of “us”) have attempted here on Earth: parachute extraction of battle tanks from aircraft, followed by parachute descent, and last-second rocket braking to a touchdown, parachute still attached. The Russians did this, somewhere around 1960. It does offer a way to make aero-decelerators more effective, at the expense of extra rocket fuel and hardware.

Here’s another thought: augment the hypersonic aero-braking with rocket retro thrust during atmospheric entry. Once that is over, deploy aerodynamic decelerators, but maintain rocket retro thrust. Increase rocket braking thrust sharply for the final last seconds to touchdown. The main problem is axial retro plume aerodynamic stability, which can induce control moments on the craft beyond its capability to cope. Nozzle cant angle might help resolve that, however. See Figure 2 below.

There are many questions and problems with this overall retro-thrust concept, but it actually does offer a possible way to land very large payloads on Mars, assisted by aero drag retardation. See Figure 3 (added as an update 6-10-12).

Figure 1 – How Aero-Decelerators Work

Figure 2 – Stability Problems with Hypersonic Retro Thrust Might Be solved by Cant Angle

Figure 3 - Combined Chute and Rocket Braking