Acceleration

RocketCat sez

Delta-V is the key to most space flight. As long as your ship can crank out enough delta-V for the mission, you don't give a rat's heinie about your acceleration.

With two very important exceptions.

One: when blasting off, the planet has its hand out for a "gravity-tax" on your delta-V. Terra's gravity subtracts 9.81 meters per second every freaking second. If your ship cannot accelerate more than that, it's just gonna vibrate on the launch pad and slowly burn a hole in the ground. You will not go to space today. And the higher your blastoff acceleration, the shorter your lift time and the lower the total gravity-tax. Keeping in mind that it is considered bad form to kill the crew with brutal acceleration levels.

And Two: if a bad guy is shooting at you; the more acceleration, the better to dodge the missiles.

If you're not gonna eat that rat heinie, I'll take it. And pass the Sriracha sauce.

The spacecraft's acceleration depends on the total thrust and the spacecraft's mass. For most purposes, we don't care about this. The spacecraft can be theoretically any size. The equation is

A = F / Mc

where

  • A = spacecraft's acceleration (m/s), divide by 9.81 for Gs
  • F = spacecraft's thrust (newtons)
  • Mc = spacecraft's current mass (kg)
Example

If the Arcturus can manage 19,620,000 newtons of thrust and masses 200,000 kg, 19,620,000 / 200,000 = 98.1 m/s or 10 gs of acceleration.

As a short cut, you can calculate acceleration using the Transit Time Nomogram

We don't care about acceleration, that is, with the major exception of landing and take-off. If the Polaris is taking off from Terra, and it does not produce acceleration greater than 1 g, it is not going to move even a millimeter higher. For these calculations, for Mc use the spacecraft's mass with full propellant tanks. As a rule of thumb, you want the spacecraft capable of doing 1.5 g, though 1.3 g will do in theory, and 10.0 g will really reduce the gravitational drag. 1.5 g = 14.72 m/s. The value you pick will be what you will use to calculate Apg in the gravitational drag formula.

Oh, and another thing: keep the acceleration below 30g to avoid injuring the astronauts.

On the Transit Time Nomogram, the minimum liftoff values are labeled on the Acceleration scale for your convenience. In the example above, a 46 metric ton spacecraft with a particle-bed nuclear thermal propulsion system can accelerate at 0.5 g. Glancing at the chart, you can see that the spacecraft has no trouble lifting off from Mercury, Mars, and the various moons; but cannot lift off from Venus or Earth.

This means that the engine's so-called "thrust to weight ratio" has to be higher than 1.0 if the rocket is expected to take off from Terra. (You can get away with less on smaller planets. Maybe.) Sometimes you are lucky and can find this value while researching propulsion systems. Lucky you, I included this data in the engine table above. Bottom line: do not use any engine marked "no" in the T/W>1.0 column if the spacecraft has to be capable of takeoff or landing.

At this website, they suggest that the optimum thrust to weight ratio varies from 1.15 to 1.2.

Gross Lift Off Weight

By rearranging the equation for acceleration, given the ship's thrust we can calculate the maximum mass of the ship with full propellant tanks (the mass of the ship with full tanks is often called Gross Lift Off Weight or GLOW).

GLOW(kg) = Thrust(newtons) / accel(m/s)

GLOW(kg) = Thrust(newtons) / 14.72(m/s)

A single Gas Core engine has a thrust of 3,500,000 newtons. If Polaris has one GC engine, its maximum liftoff mass is 237.8 tons, which is pretty disappointing. ( 3,500,000 / 14.72 = 237,771 kg )

But if it had five GC engines, it would have a liftoff mass of 1188.9 tons. That's more like it.

This also can be calculated with the Transit Time Nomogram

Other Considerations

The other major exception is that a ship's acceleration affect maneuverability. This is important if somebody is shooting at you. It is hard to jink when your acceleration is measured in humming-bird powers.

Note that adding more engines only increases the acceleration and thrust (and the rate of propellant consumption). It does nothing to the deltaV or exhaust velocity. It also cuts into the payload mass. Also note that if an engine has a thrust to weight ratio below one, it doesn't matter how many of them you add, it still won't be able to lift-off.

Multiple engines produce other problems that have to be taken into account. If they are too close together, they inflict their waste heat on each other, increasing the heat radiator requirements. If they are too far apart and are of a type that emits nuclear radiation, they increase the number of shadow shields required, which cuts into the payload.

Think of the Earth as being at the bottom of a funnel-shaped well whose walls become less steep as you climb away from Earth. (ed note: the "gravity well")

Paint the walls of the funnel in zones of different colors to represent the various space traffic control center jurisdictions. The ones nearest Earth at the bottom of the funnel are controlled from national centers that are, you hope, in communication with one another and swapping data. The ones farther out are watched by seven other centers located in GEO. And the ones in the nearly-flat upper part of the funnel are four in number centered on L-4, the Moon, L-5, and a huge "uncontrolled sector" stretching around lunar orbit from 30-degrees ahead of L-4 to 30-degrees behind L-5 where there wasn't anything then.

Now spin the funnel so the bottom part representing a distance up to 50,000 kilometers goes around once in 24 hours. Spin the top part from 50,000 kilometers altitude out to a half-million kilometers at the lunar rate of 29.5 days.

Located on the walls of this madly turning mult-colored funnel are marbles spinning around its surface fast enough so they don't fall down the funnel. Some of them are deadly marbles; come close enough and you'll burn. Others are big and fragile, but massive enough to destroy your ship if you hit one. Still others are ships like your own, plying space for fun, profit, or military purposes. An unknown number of the last are capable of whanging you with various and sundry weapons.

Your mission: without coming afoul of any of this, get to the flat tableland on top, then locate and dock to a group of fly-specks called L-5.

Try it on your computer. Good luck.

From Manna by Lee Correy (G. Harry Stine) 1983
From Space Angel (1962). The spaceport launch pads are deep tubes, presumably to keep the radioactive engine away from the ground crew.

Remember that the ship has to be balanced around the axis of thrust or it will tumble. Cargo will have to be stowed in a balanced manner, and logged in a mass distribution schedule (sometimes called a "Center-of-mass and moment-of-inertia chart).

They were in a conical room. Above them the pilot lay in his acceleration rest. Beside them, feet in and head out, were acceleration couches for passengers. "Get in the bunks!" shouted the pilot. "Strap down."

Ten boys jostled one another to reach the couches. One hesitated. "Uh, oh, Mister!" he called out.

"Yes? Get in your couch."

"I've changed my mind. I'm not going."

The pilot used language decidedly not officerlike and turned to his control board. 'Tower! Remove passenger from number nineteen." He listened, then said, "Too late to change the flight plan. Send up mass." He shouted to the waiting boy, "What do you weigh?"

"Uh, a hundred thirty-two pounds, sir."

"One hundred and thirty-two pounds and make it fast!" He turned back to the youngster. "You better get off this base fast, for if I have to skip my take-off I'll wring your neck."

The elevator climbed into place presently and three cadets poured across. Two were carrying sandbags, one had five lead weights. They strapped the sandbags to the' vacant couch, and clamped the weights to its sides. "One thirty-two mass," announced one of the cadets.

"Get going," snapped the pilot and turned back to the board.

Matt pulled himself along, last in line, and found the scooter loaded. He could not find a place; the passenger racks were filled with space-suited cadets, busy strapping down.

The cadet pilot beckoned to him. Matt picked his way forward and touched helmets. "Mister," said the oldster, "can you read instruments?"

Guessing that he referred only to the simple instrument panel of a scooter, Matt answered, "Yes, sir."

"Then get in the co-pilot's chair. What's your mass?"

"Two eighty-seven, sir," Matt answered, giving the combined mass, in pounds, of himself and his suit with all its equipment. Matt strapped down, then looked around, trying to locate Tex and Oscar. He was feeling very important, even though a scooter requires a co-pilot about as much as a hog needs a spare tail.

The oldster entered Mart's mass on his center-of-gravity and moment-of-inertia chart, stared at it thoughtfully and said to Matt, "Tell Gee-three to swap places with Bee-two."

Matt switched on his walky-talky and gave the order. There was a scramble while a heavy-set youngster changed seats with a smaller cadet. The pilot gave a high sign to the cadet manning the hangar pocket; the scooter and its launching cradle swung out of the pocket, pushed by power-driven lazy tongs.

A scooter is a passenger rocket reduced to its simplest terms and has been described as a hat rack with an outboard motor. It operates only in empty space and does not have to be streamlined.

The rocket motor is unenclosed. Around it is a tier of light metal supports, the passenger rack. There is no "ship" in the sense of a hull, airtight compartments, etc. The passengers just belt themselves to the rack and let the rocket motor scoot them along.

From SPACE CADET by Robert Heinlein. 1948

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