A report of the stall speed of a Cessna 206 on Mars

[Cadet_BNSF]

A friend and I were in our English class and were bored. So my friend, @RisingAeroEngineer11, posed this question to me: What is the stall speed of a Cessna 206 on Mars?

I initially thought that it could only be experimentally determined, but after consulting the oracle (Google), we came up with this equation

V = √( 2 M g / ρ S C_{l max} ) 

Where

V = stall velocity

M = mass

g = acceleration due to gravity

ρ = atmospheric density

S = wing area

and 

C_{l max} = Coefficient of lift, max

Finding the  variables related to the aircraft itself were surprisingly hard to find. The mass was easy with the mass of a Cessna 206 with 65 gallon tanks and full fuel, assuming a 70 kg pilot, being 1,234 kg. Now, in actuality, a combustion powered plane could not run on Mars, so we will assume that the engine is replaced by an electric motor of the same weight and the fuel tanks are replaced with batteries of the same weight, however unrealistic that may be. After looking at a couple of websites, we eventually found the wing area, which is 16.3 m². The hardest variable to find was actually the Coefficient of lift, which we actually had to solve for.

To find the coefficient of lift we solved the stall speed equation on Earth, with all of the variables plugged in. The published stall speed of a Cessna 206 is 52 knots, which is 26.75 m/s.

So the equation looked something like this:

 

 

Rearranging to solve for C gives an answer of 1.694.

Now that we know all of our plane's variables, we can plug them into the equation. We used a gravitational acceleration of 3.711 m/s²  and an air density of 0.02 kg/m³.

When everything is plugged in, the equation looks like this

 

 

 

When all is said and done, this gives us a velocity of 128.78 m/s, or 286 mph.

Now, a quick google search tells us that the local speed of sound is 240 m/s, so in order to just take off, our 206 has to go almost half the speed of sound!

Thanks for giving me your time, and comment below if we made any mistakes.

 

[Geonovast]

As always...

https://what-if.xkcd.com/30/

 

 

[RisingAeroEngineer11]

Hah! Yup, had echoes in my head of that awesome XKCD comic while we were working. But wouldn’t you agree running the numbers is more fun? Kudos to @Cadet_BNSF for creating this post. 

[PB666]

2 hours ago, Geonovast said:

As always...

https://what-if.xkcd.com/30/

 

 

Technically speaking if you were on Jupiter you could carry oxygen instead of fuel and run off of hydrogen in the jovian atmosphere. Instead of a piston propelled aircraft you would use a jet turbine capable of producing three times the thrust then vector some of that down to stay aloft longer.

Of course a cessna flying on Mars could have and would need a larger wing area to reduce the stall speed.

There is no such thing as a 'generic' stall speed. What they mean is a level-flight stall speed. A stall occurs when your angle attack exceeds the maximum allowed and maintain bernolli's lift, this is about 15 degrees above the direction of motion. This can occur at any speed even Mach Speed (especially at higher altitudes).  This means that for an airplane lift continues to increase to about 12' AoA and then life flattens out and then suddenly falls. As lift falls drag markedly increases.

For level flight the slower you go the more AoA you need to  maintain level flight (and the more drag you produce) however once you reach a speed in which AoA is about 12 you should not further lower speed.

You can do some fun things with stalls though, on flight simulator I pulled a 777 into a stall about 15 miles from runway at 45,000 feet (rather easy to do at that altitude because you are in coffins corner) and began tossing out flaps spoilers and gear, then dove the nose down to take advantage of the drag that the stall created holding the nose ever so slightly above the stall angle of attack, when you get close to the glide slope you remove the spoilers and roll the nose down below the horizon and add landing thrust.

Of course if it was real life the 777 would have flipped over, because there is not enough laminar air flow over the flight control surfaces to maintain stable flight.

 

[PakledHostage]

2 hours ago, PB666 said:

Of course if it was real life the 777 would have flipped over, because there is not enough laminar air flow over the flight control surfaces to maintain stable flight.

 

If it was real life, the 777's primary flight computers would have commanded the elevators to pitch the aircraft nose down.

[KG3]

How about the stall speed of a Cessna 206 on Titan?

[Green Baron]

Guys don't you think that, if someone wanted one day to fly a rigid wing contraption on Mars, it would have a different design than a standard aircraft on earth ?

[PakledHostage]

4 minutes ago, Green Baron said:

Guys don't you think that, if someone wanted one day to fly a rigid wing contraption on Mars, it would have a different design than a standard aircraft on earth ?

Yes.

 

1 hour ago, KG3 said:

How about the stall speed of a Cessna 206 on Titan?

See Geonovast's post above.

[Green Baron]

Just now, PakledHostage said:

Yes.

My faith in my colleagues returns :-)

[Cadet_BNSF]

15 hours ago, PB666 said:

Technically speaking if you were on Jupiter you could carry oxygen instead of fuel and run off of hydrogen in the jovian atmosphere. Instead of a piston propelled aircraft you would use a jet turbine capable of producing three times the thrust then vector some of that down to stay aloft longer.

Of course a cessna flying on Mars could have and would need a larger wing area to reduce the stall speed.

There is no such thing as a 'generic' stall speed. What they mean is a level-flight stall speed. A stall occurs when your angle attack exceeds the maximum allowed and maintain bernolli's lift, this is about 15 degrees above the direction of motion. This can occur at any speed even Mach Speed (especially at higher altitudes).  This means that for an airplane lift continues to increase to about 12' AoA and then life flattens out and then suddenly falls. As lift falls drag markedly increases.

For level flight the slower you go the more AoA you need to  maintain level flight (and the more drag you produce) however once you reach a speed in which AoA is about 12 you should not further lower speed.

You can do some fun things with stalls though, on flight simulator I pulled a 777 into a stall about 15 miles from runway at 45,000 feet (rather easy to do at that altitude because you are in coffins corner) and began tossing out flaps spoilers and gear, then dove the nose down to take advantage of the drag that the stall created holding the nose ever so slightly above the stall angle of attack, when you get close to the glide slope you remove the spoilers and roll the nose down below the horizon and add landing thrust.

Of course if it was real life the 777 would have flipped over, because there is not enough laminar air flow over the flight control surfaces to maintain stable flight.

 

True. I guess this is more of the takeoff speed that you would want to aim for. 

Whatever the case, it was fun to do!

[KerikBalm]

I would go about it differently to get a rough idea of the stall speed.

Cesna 172 (edit, oops, you're asking about the 206) stall speed on Earth: 40 knots = 74 km/h (although this was said to be fully loaded, flaps fully deployed). From the website I saw, this seems to be the value for sea level at 15 C (= 288K). *edit* Cesna 206 stall speed is 52 kts.

So what is the difference in air density?

Air density is a function of temperature, pressure, and MW. 

Maximum martian surface pressure: 1.155 kPa. Earth sea level: 101.325 kPa. This is 1.14% of Earth's Sea level pressure.

The martian surface can reach 30 C, but its average is 210 K vs the 288K value for Earth. So the average temperature is 210/288 or 0.7292x Earth's temperature.

MW is ~43 vs ~28.8, so 1.493x Earth's.

So... 0.0114 * 288/210 * 43/28.8 = 0.0233x as dense as Earth's atmosphere at the bottom of the Hellas Basin (although maybe it didn't reach that 1.15 kPa at 210K, if it was warmer this is an overestimation)

So at the same speed, the plane should generate 0.0233x as much lift. It only needs 0.376x as much lift thanks to Mars' lower gravity... so, lets normalize this to the surface gravities: At the same speed, the cesna on mars will generate only 0.0621 x as much lift relative to the surface gravity as on Earth. Speed needs to increase to have the lift generated be 1/0.0621 x higher which is 16.11 times more.

Ignoring mach effects, lift follows a v^2 relationship. So the speed increase needed is sqrt(16.11) = 4.01.

So I assume that its stall speed will be 4x higher than on Earth (at the bottom of Hellas, at 210K)

4.01* 75 km/h = 296 km/h, or 160 knots.

If we instead use the "standard" pressure for mars of 0.6 kPa instead of 1.15 kPa, this increases to ~410 km/h. (= 256 mph, which is fairly close to the OP's value, but below it *edit* adjusting for a cesna 206 not a 172: 256 * 52/40 = 332.8 mph.. above OPs value)

If we do this in the Martian summer, around noon when the temperature reaches 300k.... the stall speed should increase to about 490 km/h (306mph, again close to OP's value, but above it)

If this was done on top of olympus mons (still assuming 300k) which has a surface pressure of about 0.03 kPa (which is 1/20th of the "standard"), it would increase the stall speed by another 4.47x to a ridiculous 2191 km/h.

At this point mach effects are very very important, and the estimate is probably way off... probably for the worse. One can forget about atmospheric flight above olympus mons

Edited November 25, 2017 by KerikBalm
typos and stuff

[Zeiss Ikon]

You want to fly on Mars, you need a vehicle closer in overall design to an SR-71 than to a Cessna of any model (even a Citation).  The SR-71 routinely flew at up to 1000 m/s, at altitudes of up to 30 km on Earth, and the limiting factors on its speed were skin heating and engine power (thrust drops as speed rises, even with the engines running in ramjet mode).  Lack of a suitable air-breathing engine (due to lack of air) aside, the advantage of the SR-71 design is that the airframe is built to go really fast.  In the (roughly) 1% atmosphere on Mars, it wouldn't need to fly 10x as fast, it can simply pitch up a little and fly at higher coefficient of lift to regain much/most of what was lost.  Say, at worst, the aircraft has to fly 3x as fast (giving 9x the lift due to speed, and picking up the other 90x factor from flying at higher C_L, you're now flying at 3 km/s.  The aircraft already has a reaction control system, no need to add that -- just change out the turbo-ramjets for (much lighter -- we'll replace the extra weight with fuel) hypergolic bipropellant rockets with similar thrust.

Now, takeoff and especially landing will be a major stumbling block.  Stall speed calculations work the same way, so you'd need to land at a (rather hot, by any standard) approximate 700 m/s (yes, that's supersonic on Mars), and your drag parachutes won't be worth much at all.  Better install some small braking engines where the engine intake cones used to be...

[KerikBalm]

Well, I just realized I was talking about a Cesna 172, when the OP mentioned a Cesna 206... the Cesna 206 has a higher stall speed than the 172, so that explains some differences as well.

1 hour ago, Zeiss Ikon said:

You want to fly on Mars, you need a vehicle closer in overall design to an SR-71 than to a Cessna of any model (even a Citation).  The SR-71 routinely flew at up to 1000 m/s, at altitudes of up to 30 km on Earth, and the limiting factors on its speed were skin heating and engine power (thrust drops as speed rises, even with the engines running in ramjet mode).  Lack of a suitable air-breathing engine (due to lack of air) aside, the advantage of the SR-71 design is that the airframe is built to go really fast.  In the (roughly) 1% atmosphere on Mars, it wouldn't need to fly 10x as fast, it can simply pitch up a little and fly at higher coefficient of lift to regain much/most of what was lost.

Well, when looking at an airplane design to use on Mars, we should look at high altitude designs on Earth. High altitude designs are meant to operate where the air is thin - which it is on Mars.

The SR-71 is certainly one of those high altitude designs, but landing it would be very very problematic. If it needs to fly at very high speeds to fly in thin air, then for mars it needs to land at those same very high speeds. Also I'd take a bit of issue with your comments about 10x as fast and higher coeff of lift... but I'll leave that alone for now.

So what are some other high altitude designs we can look at?

The U-2... a good starting point

But then there is also this:

https://en.wikipedia.org/wiki/Helios_Prototype

Now... I also wondered... what is the air pressue at 80,000 feet, which the SR-71 reached, and the U2 may have reached:

https://www.digitaldutch.com/atmoscalc/

... 0.0272537 atmospheres apparently... and a temperature of 220 K which is similar to the martian average of 210 K. So I'll leave out the temperature compensation, and just adjust by MW of the gas... We need an altitude where the surface pressure is 0.0172 to be equivalent to the air density at the bottom of the Hellas Basin (assuming identical temperatures). This apparently occurs right around 90,000 feet. So, a plane that can fly that high, would be able to fly on Mars if its propulsion system worked there. At this point I'd mention the possibility of using air augmented rockets/turbo-rockets/ramrockets... but electric fans also work.

That Helios solar plane is actually our best contender, doing even better than the SR-71 (at least as far as what is publicly known about SR-71 performance)

https://en.wikipedia.org/wiki/Helios_Prototype#Records

Quote

On August 13, 2001,^[1] the Helios Prototype piloted remotely by Greg Kendall reached an altitude of 96,863 feet (29,524 m), a world record for sustained horizontal flight by a winged aircraft.^[4] The altitude reached was more than 11,000 feet (3,400 m) — or more than 2 miles (3.2 km) — above the previous altitude record for sustained flight by a winged aircraft. In addition, the aircraft spent more than 40 minutes above 96,000 feet (29,000 m).

Of course, this thing was Solar-Electric powered, so it should be able to fly on Mars. It wouldn't be able to sustain flight for days on end as it does on Earth because it would receive only about half the amount of solar power, but it should be able to fly at the bottom of Hellas when its cold. - I suppose we can imagine it charging the batteries during the day when it gets too warm to fly, and then flying at night when the air is colder and denser.

 

 

[Zeiss Ikon]

Don't forget coffin corner, though.  Speed of sound on Mars isn't much above stall speed for those Cessna models -- and that's at ground level, even in the Hellas Basin.  Even a gossamer craft like the Helios would reach Mach conditions barely above stall after climbing a kilometer or two.  That was the ultimate limiter for altitude of the U-2 -- the point at which stall speed approached the airframe's Mach limit.

Lowering stall speed (Helios had a stall speed under 10 m/s) only helps so much -- by the time you're high enough to clear terrain (even avoiding the entire area of Olympus Mons), you're in coffin corner with any Mach-limited airframe (not to mention that Mars routinely has wind velocities that could crumple Helios like a used tissue).

Oh, and batteries don't like cold weather.  Even lithium batteries lose a lot of performance when cell temperature approaches 0 F (current electric and hybrid cars actually use the battery's own power to keep it warm if off charge during very cold conditions).  This lowers the effective capacity of the battery, of course, though once the motors are drawing current for flight, there's enough byproduct heat that additional energy needn't be spent on warming.

[KerikBalm]

Yes, the U-2 would not cut it. Its service ceiling was not high enough, but the design isn't a bad place to start looking. Battery output may not be so great in cold weather, but PV systems do perform better in the cold vs heat... too bad the air would thin at noon in the summer, but there would be plenty of cases where there is direct sunlight and its not 30 C (but it heats up quickly, the thermal inertia is quite low with that little atmosphere).

Any sort of aerodynamic flight poses lots of problems on Mars. A "gossamer" craft like the helios would likely not do well in the martian storms (it also broke up due to weather here on Earth), I just think that of the designs that have actually flown on Earth, it is the closest to something that would work on Mars.

It clearly still has problems, but any aircraft on Mars is going to have major problems.

 

[KerikBalm]

Ooops... I forgot (even though I mentioned it in my first post) - we don't need to find an aircraft that flies in air of the same density as on Mars, because of the low gravity of Mars.

Mars has .376 G, so a wing needs only to generate 0.376x as much lift, and it should thus be able to fly in air 0.376x as thick as it can on Earth. So the reference pressure of 1.15 kPa that I was using for the bottom of Hellas and equal to 90,000 feet on Earth, should in fact be 0.44 kPa, which is higher than the "datum" pressure of 0.6 kPa on Mars. An aircraft that can fly at 90,000 feet on Earth could fly over most of Mars. Launching and landing would be a major problem. Lower gravity helps with stall speed, but it doesn't change inertia. Those electric motors on the Helios (for example) would produce very little thrust. The plane would accelerate very slowly, and it would have a very very long takeoff run. Planes on Earth at least have the advantage of being able to push on dense atmosphere during takeoff. While the Helios could fly on Mars, and landing speed would still be very high, takeoff would be even more problematic.

Maybe a RATO boost? The Helios was designed to stay aloft indefinitely on Earth... if it could do the same on mars that would get rid of the landing problem (just don't land). The problem is again the lower solar flux. Its also that the aircraft would descend to ~50,000 feet on Earth, which is 0.11 atmospheres. Even adjusting for gravity and MW, this would still correspond to a pressure of 2.77 kPa on Mars - which is more than double the pressure at the bottom of the Hellas Basin.

I don't see a way around it, a Helios like craft would have to land after less than 24 hours of flying. So a mission where it is somehow assembled on the surface, RATO launched, and then just stays aloft the rest of the mission would be a pretty short mission. Maybe if it also had retrorockets for landing, and it carried enough fuel for X launches and landings?

Some sort of ground base with a net to catch it and a catapult to launch it (like a mini CATOBAR carrier?)

Find a very flat place, and send a rover to comb/clear away all the rocks and boulders to make a landing strip for it?

It would probably be torn up in the first martian dust storm anyway....

[Green Baron]

Hehe. That took some time to notice, @KerikBalm :-)

For a design proposal one could have a look at stratosphere craft proposals for earth, like the mentioned Helios. Slow flying, ludicrously high aspect ratio gliders might work to some extent and altitude. Could they carry anything more than their own weight ? Idk.

[James Kerman]

1 hour ago, Green Baron said:

 For a design proposal one could have a look at stratosphere craft proposals for earth, like the mentioned Helios. Slow flying, ludicrously high aspect ratio gliders might work to some extent and altitude. Could they carry anything more than their own weight ? Idk.

Airbus's Perlan 2 It's a 2 man sailplane with experiments onboard.

Quote

Engineless Perlan 2 reaches 52,172 feet, breaking record set by Perlan 1 in 2006

El Calafate, Argentina, September 4, 2017 – Airbus Perlan Mission II, the world’s first initiative to send an engineless aircraft to the edge of space, made history yesterday in the Patagonia region of Argentina by soaring to over 52,000 feet and setting a new world altitude record for gliding.

An in depth article here:
https://www.airspacemag.com/flight-today/sailplane-stratosphere-180959154/

[Green Baron]

50.000ft riding the wave, incredible. That's flying ! Yes, something like that i would think.

 

You can't fly an SR71 on Mars, it is (was ? Since 20 years retired wikipedia says ...) a highly specialized craft with oxygen breathing engines and a huge industrial apparatus to keep a few flying. Much too heavy and wrong propulsion system. Even on earth it needsed refueling to climb to a height it would have to take off from on Mars.

Edited November 25, 2017 by Green Baron

[KerikBalm]

If you replaced the Turboramjets on the SR-71 and put in air augmented rockets/ramrockets/air-turborockets, you should be able to fly it or something very close to it. IIRC the oxidizer to fuel ratio for Kerosene (which is essentially what the SR-71 used), is something like 2.6 or 2.7 to 1 - quite a bit different from the 1.22:1 ratio that we have in KSP.

So...fuel consumption would increase by something like 3.6x (2.6 tons of oxidizer per 1 ton of fuel, 3.6 tons total per 1 ton of the old engine). IIRC, the SR-71's engines actually only got about 2080 Isp. So 2080/3.6 ... I would expect trying to run the engines by just injecting oxidizer in with the fuel would get you about 570 Isp... which isn't great, but its better than a normal Hydrolox rocket. The J58 of the SR-71 was an engine of compromises though, since its a turbojet that at high speeds has a bypass/bleed system that makes it sort of a partial ramjet. I'm sure a dedicated ramrocket would get better than 570 Isp... solid fuel ramrockets have already demonstrated 500+ Isp here on Earth.

There'd be no significant thrust augmentation until higher mach numbers (around mach 0.5 I seem to recall reading), so it would have to launch as just a normal Kerlox rocket getting Isps in the high 200's/low 300's.

Mar's orbital velocity is relatively low... I wonder if a SR-71 derived plane with Ramrocket engines getting ~600 atmospheric Isp could retroburn, and fly around for a while with some air augmentation.

Also, FYI, one major reason that SR-71s had to refuel just to get to altitude is because of the temperature that they were expected to operate at, and thermal explansion. Supposedly the aircraft leaked a lot when it was on the ground. Apparently the leaks didn't stop until it got up to speed and things heated up and expanded. As a result, it didn't take off with full fuel anyway.

[Green Baron]

Question (am too lazy to calculate): what's the dV-capacity of that proposal ? Would it suffice to get off the runway, assuming tank sizes of the earthly design ? Mars' curvature might help if the tyres survive the take off run :-)

 

Nay, since we cannot even get such a thing and the industry to operate it to Mars i think the extreme light weight, high aspect ratio, high lift wings with an instrument compartment are would surely be the better choice. But, really, this is all a play of mind for me ... i leave the discussion again :-)

 

[KerikBalm]

Hmm, what would the dV be... Lets assume 600 Isp for an air augmented rocket with kerosene fuel... lets pretend that mass doesn't change

69 tons loaded, 30.6 empty. 9.81 * 600 *ln (69/30.6) = 4786 m/s ... not very good*. Also that seems to mean that the real SR-71 gets 16,620 m/s when its full and going at mach 3 or so.

Hydrolox would give better performance, but the SR-71 would need really extensive changes to carry significant amounts of cryogenic H2.

I fully agree that "extreme light weight, high aspect ratio, high lift wings with an instrument compartment are would surely be the better choice".. but a ramrocket SR-71 would be fun to think about too.

* but perhaps enough to SSTO on Mars, since its orbital velocity is around 3 km/sec, and a ramrocket can also work in a vacuum as just a normal rocket.

 

Edited November 25, 2017 by KerikBalm