Dr. Desai's Challenge on Martian Atmosphere Models.

NASA'S DR. PRASUN DESAI'S ATMOSPHERIC MODEL QUESTION. Dr. Desai assumes that because the last four landers on Mars landed downrange, the atmospheric density profile must be less than predicted. This, of course, is the opposite of what all my research about the Martian atmosphere indicates.  Let's first examine his specific question, and then look at two components of the entry, descent and landing problem. My emphasis will be to address issues related to parachute size, and then to look to the issue of atmospheric density between 10 and 50 km in light of new evidence that there is 10 to 100 times more water vapor in the Martian atmosphere than envisioned when Dr. Desai posed his question. 

 THE SPECIFIC DESAI QUESTION (see his article summary): "Does the fact that every one of these entries encountered a lower atmospheric density profile than predicted indicate a random occurrence or is there a systemic bias in current Mars atmospheric models? As such, a question is posed to the atmospheric community to consider if the Mars modeling assumptions are appropriate or are there underlying modeling issues that need to be reexamined or reevaluated. Additionally, although the entire density profile is necessary for entry, descent, and landing design; nearly all deceleration during entry occurs between 10-50 km. As such, prediction of density within this altitude band is most critical for entry flight dynamics and design.”

SIZE OF PARACHUTES USED ON MARS.

Obviously, a parachute could not be used to slow a rocket landing on the Moon.  There would be no air for the parachute to catch.  One would think that the low atmospheric pressures on Mars would require an enormous parachute to land there.  The actual landers on Mars cut the parachutes loose before landing, going to retrorockets or air bags, but the parachutes used are actually not very large.  In fact, the one used for Phoenix was reduced to 39 feet (see Figure 1 upper left Idaho test insert below) from the 42 feet used for Pathfinder. However, the rest of Figure 1 appears to show a considerably smaller parachute that Wikipedia describes as having an apparent diameter of only 10 meters (~33 feet).

I asked Dr. Desai about the parachute.  His response was:

As for parachute size, we all would like to use larger parachutes. However, as the volume of these landers is small, there just is not sufficient room to fit a larger parachute. We have to make tradeoffs on various systems in order to arrive at a design that meets overall requirements. That is also why there typically is some sort of a retro-rockets system prior to landing to decrease the velocity further prior to landing. It all comes down to systems design of the lander and the tradeoffs that are necessary. The other aspect of the parachutes is that their impact is an integrated effect. As such, throughout the descent after they are open, the length of time is what makes their use worthwhile even with the low atmospheric

      What would happen in terms of the parachute if the air was less dense than planned for by the NASA EDL team? If less dense all the way down, the parachute would be less effective, and the probe would come in faster and possibly crash.  However, we have seen above that Desai admits the density is higher down low when it is lower up high.  Somehow the normal scale height ratios are not holding on Mars, but this begs the question of why they do not hold?  Are there radical alterations of density profiles that are causing probes to land long or crash?

      If the atmosphere was more dense than planned for down low, the parachute would work, but the probe would take longer to reach the surface, perhaps drifting for more time in the Martian winds.  It would land long, as was the case with Pathfinder, Spirit, Opportunity, and Phoenix.

PROBLEMS IN IDENTIFYING PARACHUTE SIZE. If there seems to be some confusion about which parachute diameter to use for Phoenix calculations, consider the article entitled Mars Exploration Entry, Descent and Landing Challenges by Dr. Robert D. Braun of the Georgia Institute of Technology and Robert M. Manning of the Jet Propulsion Laboratory at the California Institute of Technology. Table 1 of their 2005 paper lists the (planned) diameter of the Phoenix DGB (Disk-Gap-Band) Parachute as 11.5 meters (37.73 feet). The same table lists the DGB parachute diameter of the Viking 1 and Viking 2 as 16 m, but on Figure 12 the diameter shown is only 11.7 m. For Mars Pathfinder Table 1 lists the DGB parachute diameter as 12.5 m, but Figure 12 shows its diameter as 7.9 m. Table 1 further lists the DGB parachute diameter for Mars Exploration Rovers A (Spirit) and B (Opportunity) as 14 m, but Figure 12 shows just 8.9m. It looks like the different diameters may be due to differences in how the diameters are measured. Probably the larger figures represent the diameter of the 2D parachute when spread out on the floor, while the smaller diameters represents the size seen when the edges are pulled in due to the attached suspension lines, riser and payload and below the canopy and skirt.

       The Braun and Manning paper indicates that all parachutes used for Mars landings including that planned for the Mars Science Lab (currently enroute to Mars) were flight tested only in the 1960s. Due to great expense, these tests were not ever repeated. To see one such test of a 40-foot (12.192 m) parachute, click here. 

Original Caption Released with Figure 1 Image (minus Idaho test insert):

Mars Reconnaissance Orbiter's High Resolution Imaging Science Experiment (HiRISE) camera acquired this image of Phoenix hanging from its parachute as it descended to the Martian surface. Shown here is a 10 kilometer (6 mile) diameter crater informally called "Heimdall," and an improved full-resolution image of the parachute and lander. Although it appears that Phoenix is descending into the crater, it is actually about 20 kilometers (about 12 miles) in front of the crater.

The Phoenix Mission is led by the University of Arizona, Tucson, on behalf of NASA. Project management of the mission is by NASA's Jet Propulsion Laboratory, Pasadena, Calif. Spacecraft development is by Lockheed Martin Space Systems, Denver.

Image Credit:

NASA/JPL-Caltech/University of Arizona

Image Addition Date:

2008-05-27

CHECKING NASA'S MATH. There were several problems with the math offered by NASA for the EDL. One is that there are different diameters offered online for the parachute, another is that the HIRISE photo seems to suggest even an lower diameter, and finally there is the issue of the late parachute deployment.          

       I first calculated an estimate of the velocity of the Phoenix Lander during the parachute deployment phase using the largest parachute diameter published (39 feet = 11.8872 m).  Because NASA published certain numbers (given below) I wanted to test them. So I solved the following equations numerically:

d²z/dt² = -g + ρ(z)*(dz/dt)²*A*sin(θ)/m

and

d²x/dt² = ρ(z)*(dz/dt)²*A*cos(θ)/m

       It appears that at the end of the simulation, where the altitude was 925 m, the velocity was -53.605 m/s.  This agrees with the 52-54 m/s given in the Statistical Entry, Descent and Landing Performance Reconstruction of the Mars Phoenix Lander paper by Dutta et al.  If the surface air density is increased by a factor of 10, the velocity at the point where the parachute is ditched is -16.6733 m/s instead.  The reason for this test is to see what effect higher air density would have on the velocity profile.  Why a factor of 10?  There was a recent article about there being 10 times more water vapor in the atmosphere than previously known.  If this refers to a partial pressure I should multiply the surface pressure by this factor.  There is a linear relationship between pressure and density of a gas at a constant temperature.  So as a rough estimated the density was multiplied by a factor of 10. 

WHICH PARACHUTE DIAMETER IS CORRECT? The parachute diameter is uncertain by about 1.882 m because different NASA sources put it at different numbers. Most sources listed the diameter as 39 feet (11.882 m), while a few used 10 m (32.8 feet). There was a 6.6 second delay in the parachute deployment; therefore I will have to solve the 20-50 km part of the trajectory later.  I believe this is due to activity in this part of the atmosphere or an engineering flaw. Dr. Desai states that  most deceleration occurs below 50 km, but he uses different altitudes at different times for where this deceleration largely decreases.  The middle atmospheric part will have varying scale heights as the distance traveled is larger and there is supposed to be an inversion layer somewhere.  This means the density profile will be piecewise. Note that this parachute program is a 2-D model, but I don't predict how far downrange a craft will go.  The MATLAB code is below.

clc
clear
dt = .005;
rho0 = .02;
H = 10800;
A = pi*(5.9437)^2; %5.9437
m = 350;
z0 = 13300;
g = 3.71;
theta0 = 4.9*pi/180; %13.3
v0z = -387.6*sin(theta0);
v0x = -387.6*cos(theta0);
velocityz = [];
velocityx = [];
velocity = [];
altitude = [];
iter = 0;
while z0 > 925
    rho = rho0*exp(-z0/H); %Scale Height gives density
    z = z0 + v0z*dt; %Vertical position
    vz = dt*(-g + (rho*v0z^2*A/m)*sin(theta0)) + v0z; %Vertical velocity
    z0 = z;
    v0z = vz;
    vx = dt*(rho*v0x^2*A/m)*cos(theta0) + v0x; %Horizontal velocity
    v0x = vx;
    altitude = [altitude,z];
    velocityz = [velocityz,vz];
    velocityx = [velocityx,vx];
    iter = iter + 1;
    velocity = [velocity, sqrt(velocityz(iter)^2 + velocityx(iter)^2)];
    disp(z)
end
plot(velocity,altitude)

PROBLEM: The parachute deployed 6.6 seconds late. The next invesitgation will examine the issue  of whether 10 to 100 times more water vapor in the atmosphere than expected accounts for this delay via increased buoyancy in the atmosphere. I first calculate an estimate of the velocity profile from 13 to 50 km of the Phoenix lander (later I will focus on the 13 to 40 km fragment which I believe will most closely explain the 6.6 second delay).       

       Once again the forces were due to gravity and ram pressure that is a function of density. Varying scale height profiles were used to attempt to best mimic the actual atmospheric density profile.  So the equation is once again:

d²z/dt² = -g + ρ(z)*(dz/dt)²*A*sin(θ)/m

       Where the density ρ is given by: ρ = ρ₀*exp(-z/H).  The surface air density ρ₀ is .02 kg/m³.  Since “H” is a function of temperature, a temperature profile was used.  “A” is the cross-section of the space craft, which is 5.47 m².  The angle is the initial entry angle and is assumed constant (13.3 degrees).  The Martian gravity “g” is 3.711 m/s² and the initial altitude and velocity were 50 km and 1180 m/s (downward component) respectively.  As for the total velocity, it is 5450 m/s, which was taken from a paper by Withers and Catling (2010).  The lander mass “m” was 582 kg.  This problem was broken into two ODE’s (Ordinary Differential Equations) and solved as initial value problems with error in the second order (I used a time step of .0005 s to attempt to compensate). 

       The first run assumed the density at 56 km was the given 10^-4 kg/m3 (see Figure 7).  When run like this, the downward velocity at 13.3 km altitude was 448 m/s and it took 44.03s to transverse this distance.  However, using this offical figures yielded a surface pressure of  23-24 mb.  The next test asked what density at an altitude of 56 km causes the surface pressure to be ~8-9 mb.  When run like this, the downward velocity at 13.3 km altitude was 880 m/s and it took 37.22s to transverse this distance.  This velocity is not the ~387 m/s measured on the craft, but it does account for the ~6.6 s.  Note the code below can be altered by changing rho0 before the loops.  While it appears the code works, it may be changed later if I find bugs.

format 'long'

dz = 10; %Height Increment

M = 44*(1.66*10^-27);

k = 1.38*10^-23;

g = 3.71;

rho0 = 10^-4;

z0 = 56000-dz;

z = 56000; %Used to compute height difference

P = []; %Stores Pressures.  Pressure is due to the weight of the atmosphere

%above.

while z0 > 50000

    T = 2.25*10^-3*(54000-z0) + 145; %Compute Temperature

    H = k*T/(M*g); %Compute Scale Height

    rho = rho0*exp((z-z0)/H); %Compute Density using z as baseline height

    %and z0 as the current height.  This way dz is always the height

    %difference used in the calculation.

    z = z0; %Set the upper bound of altitude as the current lower bound

    z0 = z0 - dz; %Set the lower bound of altitude as current minus dz.

    rho0 = rho; %Set the current density as the baseline density

    P = [P,rho*g*dz];

end

while 50000 >= z0 && z0 > 35000

    T = 153;

    H = k*T/(M*g);

    rho = rho0*exp((z-z0)/H);

    z = z0;

    z0 = z0 - dz;

    rho0 = rho;

    P = [P,rho*g*dz];

end

while 35000 >= z0 && z0 > 0

    T = 2*10^-3*(35000-z0) + 153;

    H = k*T/(M*g);

    rho = rho0*exp((z-z0)/H);

    z = z0;

    z0 = z0 - dz;

    rho0 = rho;

    P = [P,rho*g*dz];

end

disp(sum(P)/100) %Sum all pressure intervals to get surface pressure (mb)