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At Perseverance's and Ingenuity's location the atmospheric pressure is about 750 Pa (0.11 psi). On Earth, we find that pressure about 110,000 ft high. If we built a copy of Ingenuity, could we bring it up there or close to that altitude? Probably not, because of the Earth's higher gravity and its atmosphere's lower density, but how high could we fly it?

The highest vertical rotorcraft so far went to "only" 40,800 ft (12.4 km), this was a massive crewed helicopter.

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    $\begingroup$ You’re right - although the atmospheric pressure is the same at that altitude, the gravity is significantly higher and so the helicopter would not be able to fly. $\endgroup$
    – Frog
    May 17, 2021 at 11:02
  • $\begingroup$ I'm pretty shure it couldn't get that high on its own, starting from the ground, because of the limited battery capacity. But still it would be interesting to know how much density it would need to fly in earth's gravity. $\endgroup$
    – Ulli T
    May 17, 2021 at 11:10
  • $\begingroup$ @UlliT But let's suppose Ingenuity's copy is solar-powered. $\endgroup$
    – Giovanni
    May 17, 2021 at 12:04
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    $\begingroup$ @Giovanni Not sure adding mass in the form of solar panels would make things better. Ingenuity is very heavily optimized for low mass. They literally looked for every gram to be removed from the total. If you make such drastic changes you would basically have an entirely different vehicle altogether. I'm also not sure where you would put solar cells on a vehicle like that. $\endgroup$
    – Ulli T
    May 17, 2021 at 13:00
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    $\begingroup$ In the first half of the video, it's not really flying, it's more like hopping up and then crashing back down. In the second half of the video, it is suspended from a cable that takes off 2/3rds of the weight in order to simulate Mars gravity. $\endgroup$
    – Jörg W Mittag
    May 17, 2021 at 18:10

1 Answer 1

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Using the simple momentum theory, the thrust generated by a rotor is:

$ T = \sqrt[3]{2 \rho A P^2} $

where $A$ is the rotor area, $\rho$ is the density and $P$ the power needed to spin the rotor. Since Ingenuity uses two coaxial contra-rotating rotors, this value has to be doubled:

$ T = 2 \cdot \sqrt[3]{2 \rho A P^2} $

The highest thrust is normally required for hovering where it simply equals the weight:

$ T = W = mg $

where:

  • $m$ is the mass
  • $g$ is the gravity

As a proof, let's introduce on the right part of the equation the values for Ingenuity and let's see if we get the correct thrust i.e. weight on the left. So:

  • $\rho = 0.01 kg/m^3$ Note: density on Mars changes according to season from 0.01 to 0.02; we use the lowest value in order to be conservative
  • $A = \pi \cdot 0.6^2 m^2$
  • $P = 350W$

Considering the martian gravity of $g_{mars} = 3.72m/s^2$ this returns a mass of $1.88kg$ which matches quite perfectly the official mass of $1.8kg$. So, even on Mars, Earth's physics is right 😅

Now, according to the question we have to bring Ingenuity on Earth at some $30km$ height and see if it can still hover. Well, the right part of the equation remains the same since at $30km$ height the density is the same as on Mars. But the left part doesn't match anymore because the gravity is now $g_{earth} = 9.5m/s^2$ i.e. some 2.5 times higher than on Mars. So no, at 30km height, gravity is too high on Earth to let Ingenuity hover.

Anyway if we go down to some $16km$ height then the density rise to $0.15kg/m^3$ giving again a perfect match between thrust and weight. Obviously other aerodynamic or structural issues have not been considered and could play an important role.

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