Astronomy & Celestial Mechanics
the sky, computed
Generator

The entry-profile generator

Deceleration against altitude, for two vehicles and three angles
Deceleration against altitude, for two vehicles and three angles. Deceleration in Earth gravities against altitude, for entries at 7.8 km/s from 122 km at 1.5°, 4°, 7° below the local horizontal, each flown at ballistic coefficients of 80 and 400 kg/m². Every curve is integrated with gravity, the curvature of the path and drag all present; nothing here is the closed form. The pairs peak at the same value and at different altitudes: 1.5° reaches 8.5 g at 51 km and 39 km; 4° reaches 13.0 g at 51 km and 39 km; 7° reaches 20.8 g at 48 km and 37 km. That is Allen and Eggers' result of 1953 and it is the reason a heat shield is designed and a load limit is not: the ballistic coefficient — mass over drag coefficient times area — decides where the vehicle is slowed and the entry angle decides how hard. Their closed form v²sinγ/2eH gives 4.1 g at 1.5°, 11.1 g at 4°, 19.3 g at 7° against the 8.5, 13.0, 20.8 integrated here — good to a few per cent where the entry is steep enough to be a straight line, and out by a factor of 2.0 at 1.5°, where the trajectory bends and the vehicle spends far longer in the air than a chord would. What the figure cannot show is lift — every trajectory here is ballistic, and a vehicle that can hold even a small lift-to-drag ratio flies the corridor rather than falling through it.

Deceleration in Earth gravities against altitude, for entries at 7.8 km/s from 122 km at 1.5°, 4°, 7° below the local horizontal, each flown at ballistic coefficients of 80 and 400 kg/m². Every curve is integrated with gravity, the curvature of the path and drag all present; nothing here is the closed form. *The pairs peak at the same value and at different altitudes*: 1.5° reaches 8.5 g at 51 km and 39 km; 4° reaches 13.0 g at 51 km and 39 km; 7° reaches 20.8 g at 48 km and 37 km. That is Allen and Eggers' result of 1953 and it is the reason a heat shield is designed and a load limit is not: the ballistic coefficient — mass over drag coefficient times area — decides *where* the vehicle is slowed and the entry angle decides *how hard*. Their closed form v²sinγ/2eH gives 4.1 g at 1.5°, 11.1 g at 4°, 19.3 g at 7° against the 8.5, 13.0, 20.8 integrated here — good to a few per cent where the entry is steep enough to be a straight line, and out by a factor of 2.0 at 1.5°, where the trajectory bends and the vehicle spends far longer in the air than a chord would. What the figure cannot show is lift — every trajectory here is ballistic, and a vehicle that can hold even a small lift-to-drag ratio flies the corridor rather than falling through it.

2 essays call entry-profile. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

A corridor 0.78° wide. Peak deceleration against entry flight-path angle, from sixty-one integrated entries at 11 km/s and β = 250 kg/m². The steep edge is where the load reaches 12 g, at 6.05°. The shallow edge is skip-out: below 5.27° the vehicle passes through the upper atmosphere and leaves again at 8.55 km/s, having lost too little speed to be captured. The corridor between them is 0.78° wide, which at an approach speed of 11 km/s is a targeting problem measured in kilometres of periapsis, days out. The curve is steep everywhere, which is the other half of the difficulty: half a degree of aiming error is a factor of 1.46 in the load. Lift is what widens this, and no ballistic capsule has any. Spaceflight

A corridor a degree and a half wide

The peak deceleration of an entering vehicle contains no property of the vehicle at all. Only the speed and the angle of arrival decide how hard it is slowed — the ballistic coefficient decides where, and nothing decides whether.

 A corridor 0.2 km wide, and a density known to a factor of 2. The apoapsis a vehicle is left on after a single atmospheric pass, against the periapsis altitude it aimed at, for ballistic coefficients of 60, 130, 300 kg/m² arriving at 3 km/s. The energy removed is the density at periapsis times an effective path length of √(2πrₚH), divided by the ballistic coefficient — so it falls exponentially with altitude and the curve is steep. Hitting a 1000 km apoapsis to ±10 per cent requires a periapsis inside 0.2 kilometres. Getting the atmosphere wrong by a factor of 2 moves the aim point by 7.6 kilometres, which is 30.7 times the corridor's own width — so a ballistic vehicle aiming at a planet whose density is known to a factor of two misses by more than the tolerance allows, and the manoeuvre has to be flown rather than aimed. Spaceflight

A manoeuvre that has never been flown once

Arrive on a hyperbola, dip once through the atmosphere, leave on a bound orbit having spent no propellant. The saving is a kilometre a second or more, the physics is the same as an entry corridor, and nobody has done it — because the corridor is a tenth of a kilometre wide and the density is known to a factor of two.

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