Mars Entry - Not So Fast !!!

• ₁ Earth total equatorial launch velocity is √(Vinf² + Vesc²) - Vrot
• ₂ pessimistic because there will be some small drag descending to circular drag
• ₃ optimistic because Apollo-style roll maneuvering reduces lift
• ₄ Apollo 16 was 4.2 peak gees

• ₅ MSL actually peaked at 12.5 gees (!!!), the mission traded lift for maneuvering and so L/D was much smaller

• gee = 9.8m/s²

The fastest orbit gets to Mars most quickly, in a fraction of an orbit time. If they miss, they need much delta V to abort and return!

NASA mars fact sheet

With all respect to my esteemed friend Gerald Norley, aerobraking in the Mars atmosphere is quite difficult.

The problem is not just the speed itself, it is the small radius and weak gravity of Mars. To stay in a thin layer of atmosphere around a small curved planet, the re-entry vehicle must make a sharp turn, which means high centrifugal acceleration. A small planet has less gravity, which somewhat counteracts the centrifugal acceleration. The remainder of the centripedal acceleration is provided by "negative lift", either wings or a lifting body shape, on temperature-limited external surfaces. Heat loading and velocity limits the "lift to drag ratio", to 1.0 for the space shuttle, 0.3 for the Apollo capsule, and 0.24 for the Mars Science Laboratory Curiousity.

Apollo entered at 11.2 km/s, essentially escape velocity. Speed creates heat, heat must be radiated and convected away.

If Apollo had entered too high up, the drag would have been insufficient for aerocapture, and it would have gone into an elliptical orbit, perhaps a slow one. The Apollo capsule had already shed the service module; the resources in the capsule would not have lasted long enough to keep the astronauts alive.

If it had entered at too steep an angle, it would have come down into the deep atmosphere with too high a velocity. The "sudden" impact would have killed the astronauts and smashed the capsule to bits.

Instead, Apollo aimed for a thin slice of atmosphere where the drag was "just right". Just right for what?

The problem was that staying in that thin atmosphere required following a circular arc at approximately the same altitude (and radius) across a considerable stretch of sky. At the onset of reentry, that required an additional "gee" of centripedal acceleration towards the center of the Earth. The only way to get that was with negative lift, with the sphere/cone shaped spacecraft cutting through the thin air and deflecting it upward, while slowing down.

The wings of an ordinary airplane creates quite a bit of lift; for every newton of drag, perhaps 10 newtons of lift, depending on the wing design and the speed. This is a lift to drag ratio ( L/D) of 10. By adding thrust with a propeller or jet or rocket, the drag can be counteracted and the lift maintained indefinitely. However, at Mach 35 orbital speeds, the wings of an ordinary airplane would be burned off or shredded to fragments. The space shuttle had strong narrow wings enclosed in heat-resistant tiles; it had a lift-to-drag ratio of about 1. A flying brick. However, those wings were heavy, and to were designed for entry speeds of less than 8 kilometers per second; if the shuttle had somehow flown in from the Moon or high orbit at 11 kilometers per second, the wings would not be strong or insulated enough to survive.

The Apollo command module had a lift-to-drag ratio of 0.3. To provide an additional gee of downwards lift, it needed 1 gee / L/D or 1 gee / 0.3 or 3.33 gees of drag to make 1 gee of downwards lift. The total force experienced by the astronauts was the vector sum of these two perpendicular accelerations, sqrt( 3.33² + 1.00² ) = sqrt( 12.11 ) = 3.48 gees. In addition, it needed some "lift" force for sideways maneuvering, so the highest gee force experienced by Apollo 16 was 4.2 gees.

Mars entry missions copy Apollo, but with an important difference; they come in faster than escape velocity. The entry velocity from Deimos is close to Mars escape velocity, about 5 kilometers per second. To stay in a circular altitude corridor around Mars at this reentry speed requires "1 Martian Gee" of downwards lift, 3.7 m/s² or 0.38 "Earth Gees". With the same L/D as Apollo, the entry acceleration would be 0.38 * sqrt( 12.11 ) = 1.32 Earth gees. Easier than Apollo!

However, entry from a slow, minimum energy Hohmann transfer orbit from Earth adds 3.55 km/s of extra velocity. In actuality, not quite that bad, because the entry velocity is the root sum squared of the velocities, or sqrt( 5² + 3.55²) or 6.1 km/s (not the naive 8.55 km/s you might expect from a linear addition of velocities. This makes sense; kinetic energy is proportional to the square of the velocity, and the total energy is the sum of the Mars infall energy and the Hohmann arrival energy.

But the results are actually worse. The centrifugal acceleration is quite a bit higher, and our calculation is more complex. The radius of Mars reentry is perhaps 3420 km, so the centrifugal acceleration is v²/r or 37.6e6/3.42e6 or 11 m/s². Subtract Mars gravity (3.7 m/s²) from that, and the required lift is 7.3 m/s², nearly double the lift of an entry at escape velocity, so the entry acceleration is nearly doubled as well, to 2.6 Earth gees. Still easier than Apollo ... although the entry profile chosen for the Mars Science Laboratory Curiosity peaked as high as 12.5 Earth gees. MSL needed more maneuvering acceleration to land in within a very small area (much tighter than Apollo); there were no helicopters and recovery fleets to rescue it if it went astray.

Similar considerations will apply to a manned Mars mission. A Mars mission designed to land within walking distance on a prior cargo delivery will need even tighter maneuvering, and will probably require powered terminal flight to do it. Without a real atmosphere, that requires rockets for hovering, horizontal translation, and landing.

Faster than Hohmann

A Hohmann transit from Earth to Mars takes 280 days. That's a lot of heavy consumables and a lot of extra lift, as well as significant biological risk from radiation and microgravity damage. 90 day transits use less consumables and reduce risk, more than compensating for the extra rocket boost needed at Earth for a faster transit. The consequences of a premature launch vehicle shutdown would be serious, and the mission would no longer get anywhere near Mars, but hopefully there would be enough delta V (intended for the return to Earth) to turn around and return home without a deadly multi-year zero-gee tour out to Jupiter and back.

Instead of an incoming "excess velocity" (or "V_infinity") of 3.55 km/s, the mission would have a V_infinity of 11 km/s. The root sum square of the velocities is 12.1 km/s, and v²/r is 146/3.42 or 42.7 m/s². Subtract Mars gravity, and the required lift is 39 m/s². The squared velocity got much bigger, but Mars did not. With the same lift-to-drag of 0.3, the drag must be 130 m/s², and the total acceleration (lift vector summed with drag) is 148.6 m/s², or 15 gees. After 90 days of weightlessness, and with added entry maneuvering gees, the result will be astronaut pudding.

This is true if we are merely slowing down to Mars orbit velocity. The problem is the high initial gees at first entry. If we want to slow to a high elliptical orbit, we will need to change v² from 146 (km/s)² to 25 (km/s)², reducing the amount of energy we must shed by only 17%. And the problem is not the total energy (though that is a problem, you need more heat shield ablation material), it is the huge acceleration at the beginning of slowdown. The first stop is a doozy.

We can go fast, and slow down to a tolerable 3 km/s for entry, merely by bringing an 8 km/s rocket along and slowing down before entry. For a Mars mission, that would be a fueled rocket about as big as a Saturn 1B. A bit heavier than a few more tonnes of oxygen and astronaut chow. Oops.

Going Fast, Entering Slow - An Alternative Brain Fart

Hold on. It's gonna get crazy.

There's another possibility. Unproved, but we can develop the technology for landing on our own Moon.

Aerogel globs launched from Deimos, precisely into the path of the heat shield of the lander, impacting in a steady dribble for DAYS before entry. Delta V from Deimos to escape is the same as its orbit velocity, 1.35 km/s, so lofting aerogel into interplanetary space requires much less delta V than bringing reaction mass fuel from Earth. If the material arrives slowly enough, in small enough chunks, the heat shield won't heat up much, far less than it would during rapid entry. Silica aerogel manufacured in full gravity on Earth can be less dense than air; what could robots make in zero gee high vacuum on Deimos?

Or in 1/6 gee high vacuum on the Moon, or in lunar orbit, or on the International Space Station, the places where we learn how to do this? Our first missions won't be Mars, but slowing down space debris, or landing on the Moon without a descent-stage rocket, or circularizing orbits at GEO.

Aiming? We could aim lasers at lunar retroreflectors in 1970. In 2017, we can deliver a shell from a Navy frigate deck gun through 40 kilometers of thick and turbulent atmosphere accurately enough to hit a dinner plate. We can measure the distance from Earth to the LAGEOS laser geodesy satellites to micrometers, and observe the effects of continental drift on the Earth stations. We can aim needles of protons moving at 0.9999999 of the speed of light within a small fraction of a micrometer to collide inside the Large Hadron Collider in Switzerland. In LIGO, we can measure mirror positions to 22 decimal place accuracy, the distance to Proxima Centauri measured to the thickness of a human hair. And of course, we can hit a 10 kilometer band of Martian atmosphere from Earth after a half-billion kilometer journey.

Oh yes, we can do superb accuracy, and we haven't even gotten started on what we will someday do with computer-controlled electromagnetic launchers (or launchloops or spinning tethers?) in vacuum, with laser distance measurement and laser-ablative course correction.

We may be able to stretch out the launch time from Deimos over months, to deliver a few hundred tonnes of aerogel in a few days of fast but gentle "pre-entry". 100 tonnes per month is 40 grams per second; at 2 km/s launch velocity from Deimos, and 50% power efficiency, that is 160 kilowatts of launch power. Solar power intensity at Deimos is half of that in Earth orbit, but zero gee mirrors can concentrate sunlight as much as necessary. Mirrors plus solar cells could mass less than a GEO communication satellite. The accelerator would weigh a lot more, but much less than the extra rocket fuel needed to slow down a high speed Mars entry.

No, I haven't done the detailed trajectory calculations. Whoever does gets naming rights; Lofstrom is too hard to spell.

All that said, the best long term way to get to Mars may be a constellation of heavy, shielded, centrifugal gravity Aldrin Cyclers.