After finding a number of calculators for parachutes, which alas had hardcoded values for atmosphere and gravity, I came upon this site. I was able to extract the formulas and create a spreadsheet using the values. The Internet has values for Martian air density and gravity, which need to be used in place of terrestrial values.
Here is the equation from the website:
Combining the two equations above for A & D leads us to the final form of the chute equation as we will use it:
D = sqrt( (8 m g) / (p r Cd v2) )
- D is the chute diameter in meters
- m is the rocket mass in kilograms
- g is the acceleration of gravity = 9.8 m/s2
- p is 3.14159265359
- r is the density of air = 1.22 kg/m3
- Cd is the drag coefficient of the chute, which is 0.75 for a parasheet (flat sheet used for a parachute, like Estes rockets), or 1.5 for a parachute (true dome-shaped chute).
- v is the speed we want at impact with the ground (3 m/s or less)
Two adjustments for Mars-gravity is 3.71 m/sec^2, and air density is .02. Plugging those into the equation, with a payload of .5kg, gives the following results:
|m is the rocket mass in kilograms||0.0623||0.5||0.5|
|cd value for parasheet or chute||0.75||1.75||1.75|
|g for earth or mars||9.8||3.71||3.71|
|rho for earth or mars||1.22||0.02||0.02|
|v for impact velocity, set to 3 m/sec or 5 m/sec||3||3||5|
|Min diameter in meters to slow to selected velocity m/sec||0.4345055384||3.8724581877||2.3234749126|
So, to slow down to a nice speed of 3 m/sec, our chute would have to be 3.87 meters in diameter. That is equal to 152 inches, or 12.6 feet. That is a rather large chute!
A slightly smaller chute from Mars Parachutes has a mass of 11.8 ounces and a diameter of 120 inches. Those convert to .334 kg and 3.048 meters. This gives a sanity check that an off the shelf parachute of about the right size to slow down the payload is available and is about the right mass. Every gram of descent chute mass is mass taken from the payload after all, so the chute needs to be as light but as strong as possible.
The Mars Balloon will need 2 parachutes.
This is a supersonic balloon, and would be designed along the lines of the parachutes used by Pathfinder and the Mars Exploration Rovers (Spirit and Opportunity). Pathfinder had a parachute diameter of 12.7 meters. This is considered the upper limit of the chute size for the balloon, as it does not have a lander per se, but the mass of helium tank etc must be included.
After the capsule has been slowed by the heat shield and the main parachute, the balloon will be inflated and the balloon will float for some period of time (planned for at least 10 days.) At some point the balloon will either lose too much gas to continue floating or the balloon envelope will rupture. In either case, the payload will descend via a secondary parachute after the cutter separates the balloon from the lower equipment. This design is very much like that employed by the terrestrial high altitude balloon designs.
Most of the high altitude designs have the parachute in line with the payload. The secondary or descent parachute does not have to slow down the payload from supersonic velocity, so can be much lighter and less rugged. The purpose of this chute is to slow the payload to the point where it can survive impact.
Preliminary estimates place the size of the descent chute to be at most 144 inches in diameter, with a mass of about .5 kg, and an appearance something like the one below.
Of special concern with the descent parachute is that it must be carried along by the balloon during the entire main mission. This imposes mass restrictions and the chute mass deducts from the science mass. Hence the diameter and materials for the descent chute must be minimized to provide for the greatest payload mass.