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 general rule, 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.

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
CONTRABAND ROCKET

(ed note: in the year 2050, our heros are members of the Southwestern Rocket Society (SRS) fan club. The fans want to travel in space in the worst way, but civilians are not allowed to fly in their own ships.

On a field trip to Luna Louis' rocket junkyard they are stunned to find the space ship Absyrtis sitting in the lot. As it turns out that ship was Mr. Louis' last command when he was in the UN Space Force, and when the ship was decommissioned he managed to obtain it at scrap metal prices.

Club president Chubb Delany has an insane idea. He tells Mr. Louis that the club would love to refurbish the old ship, and fly it on a short hop to Luna. With Mr. Louis as captain.

Mr. Louis says if the club will promise that, he will give the ship to them free, along with any used rocket parts in the lot needed for the refurbishing.)

      The day dawned bright and clear, the New Mexico sun coming up in all its splendor over the Guadalupe Mountains to the east. But the crew of the Absyrtis didn’t notice it; they had been up all night, working in the artificial daylight of floodlamps, checking and re-checking. Some of them noticed it was getting light outside, but it made no impression on them.
     The air around the ship and Luna Louis’ junk yard rang with the tension which was mounting by the minute. There were a million last-minute checks and calibrations to be made, dozens of critical items which had to be taken care of, and scores of temperamental, precise gadgets which had to be watched and watched closely.
     No one was conscious of time save for a moment in the near future that was rushing closer with every tick of the clock and announced in a booming voice over a loud-speaker.
     Luna Louis seemed calm; he wasn’t. Chubb had no time to be nervous; he was kept busy checking on the progress of the division chiefs, and the division chiefs in turn were kept busy seeing to it that every little item was taken care of as stipulated on Louis’ pre-lift check sheet. The old spaceman had compiled a complete manual—the ship’s “cook book” and the crewman’s bible—entirely from memory. No one had grounds to doubt Luna Louis’ memory; he had not been wrong in the past.
     At X-minus one hour, the passengers and their baggage were loaded. Members of the SRS who were not assigned to the crew drew straws among themselves for the berths available on this shake-down cruise. They did not make up the full load the Absyrtis was able to carry; Louis wanted to run light and put extra mass into propellants in case of trouble. And he didn’t want non-working people under foot in what was essentially a new ship.
     Greg Shearer accompanied the lawyer down to the lock, and then Luna Louis got down to businessn The final briefing was short. “I’m lifting at 5-g. I want to get out of here in a hurry to economize propellant We may need later. Don’t let the noise bother you, and don’t worry about the vibration. This bucket has a resonant structure frequency of twenty-eight cycles.
     “When you hit free fall, remember: don’t panic! If something goes wrong, you’re dead so don’t worry about it. Take your shots when we hit free fall; you must do it in case some of your men need help. Don’t use the emergency procedures unless its absolutely necessary. I trust you on that, but some of your men may get scared; that’s why each of you has a billy club under your couch pads. One man—just one man—can kill us all.”
     He stopped and look around. His eyes were now cold and hard under the visor of his brilliant red baseball cap. “We’ll sweat this first lift, all of us. But remember that the boys who really know how to do it are out there behind that fence watching us. They don't think we have a chance; I think we’ll do all right. Any questions?” There weren’t.

     Gone was the glory. Gone was the thrill. Gone was Chubb’s enthusiasm as he lay there on the co-pilot’s couch chanting off the minutes left until zero time. Sweat was rolling off him in tiny streams although Greg had long since changed to space mix which was cool as it came out of the blower duct.
     Six months ago, a space ship lift had been a wonderful thing to watch. Now, Chubb was beginning to realize it was a terribly deadly game. So much depended on so many little things, and once they were under way, there was no backing out.
     Was that stubborn solenoid valve going to stick? Had he checked that sequence circuit thoroughly enough. Suppose there was an ignition delay which could blow the tail off? Should they have gone to the time and expense of static testing the propulsion system?
     The little relays, the pieces of wire, the lengths of tubing, the bolts and nuts which he had put into this ship Without really thinking about it were now the things which stood between him and death. They had seemed so insignificant and common when he had installed them; they were something more than that now. He had known this feeling before; every engineer is inwardly stupified at the tremendous strength and power of his achievements.
     He was almost ready to call it quits, admit he was a coward, and step out of the lock. But he recalled those agonizing months of hard work refitting this old hulk of a space ship and the terrible moments when they thought they would never make it at all. And he thought of the men in the ship with him, men who had put their hearts and souls into this great adventure, had neglected then professions and deserted their friends. And there were those outside that fence who would give anything they possessed to be inside the Absyrtis at that moment.
     There was LeRoy Finch who didn’t know if his heart could stand the high accelerations. And Greg Shearer, ridden with arthritis, who forced his stiff fingers to do things that were painful to him. And there was Luna Louis.
     “Ten minutes to zero, captain!” he snapped resolutely.
     “Roger, mate! Call all hands to lift stations and report!”
     “All hands, prepare for lift! All divisions report!”
     “Electronics standing by.”
     “Power room ready!”
     “Shipmaster secure!”
     “And co-pilot and astrogation ready! All hands to lift stations and ready for lift, captain!”
     “Very well. Final checks, please. Clear our lift with Traffic!”
     “All boards, Test-Fly to TEST! Perform final checks and report compliance! Electronics, clear with Traffic Control and secure your radar contacts!”
     “Roger!”
     “Check!”
     “Right-o!”
     For the last time, Louis and Chubh ran their final checks. Item by item, they went down the list. Then Chubb said, “Final checks complete in control room, skipper!”
     “Roger,mate!” Louis switched off his intercom and spoke privately to Chubb, “It looks good, mate; it looks good. I think we might make it after all.”
     “Sure, we’ll make it, skipper,” Chubb reassured him as he started the autopilot. He watched the chronometer.
     “Five minutes to zero! Five minutes to zero! Divisions report compliance on checks”
     “Roger from power room!”
     “Roger from electronics!”
     “Roger from shipmaster!”
     “Final checks complete, skipper.”
     Louis’ voice was sharp and raspy as he spoke from his couch, flipping up the safety guards over the switches and carefully adjusting knobs, “Stand by for lift! Red light condition!"
     “Condition red, all hands! All boards, Test-Fly to FLY! Four and one-half minutes to zero!” Chubb snapped.
     “Electronics to FLY!”
     “Power room to FLY!”
     “Shipmaster to FLY!”
     “Ready, captain!”
     Louis flipped a switch. “Key your board!”
     Slipping the key from around his wrist, Chubb inserted it in his board and turned it. “Power room, you may un-lock!”
     “Un-locked in power room! Tanks pressurizing! Reactor heat coming up!”
     “Three and one-half minutes to zero!”
     “Electronics reporting! Radar forward is hunting! Green light from Traffic!”
     “If you can’t fix it, let it hunt!” Louis ordered.
     Chubb took a deep breath and threw a switch, anxiously watching tell-tale lights on the board. “Gyros uncaged and tracking! Autopilot tracking!” he reported with relief. For five hours he had babied those gyros up to speed and held them steady; it had been no easy task to erect and orient them with the ship at a tilt, and even more difficult to adjust their speed precisely so they would not precess.
     “Two minutes to zero!”
     “Steady as she goes, mate,” Louis’ voice came back levelly. “Give me thirty-second counts.” The skipper, the mastermind of the ship operation inside and out, was calm but tense. He held the reins over everything; he was the absolute master at this point, a god in a steel and titanium hull.
     “Ninety seconds to zero!”
     “Reactor to heat! Tanks pressurized! Pumps coming up!”
     “Call up your shaft speeds!” Louis requested.
     “All coming up in synch!” LeRoy’s voice boomed over the interphone. The scream of the pumps could be heard in the background. “Four thousand r-p-m—five thousand— six thousand—seven—eight—steadying—nine thousand… peaked at ninety-four hundred… They’re holding!”
     “Bearing temps and outlet pressures?” Louis was vitally interested in the performance of the pumps. They were the only mechanically moving parts in the propulsion system.
     “Normal!—Pump Five just dropped a hundred!—There it comes back!”
     They could feel it in the control room now. Those six large staged-centrifigal pumps turning over as they would shake the most solid of structures. The vibration was a piercing, pulsing scream from the deck plates, bulkheads, and overheads.
     “Sixty seconds to zero!”
     “Give me aft view on the tv monitor!” the skipper ordered. It was Bert who complied from the electronics compartment below.
     “Forty-five seconds to zero!” Chubb smoothed his coveralls under him, adjusted his panel slightly, and pulled his astrostat hood down to where he could look through its eyepiece while still keeping the panel in view.
     The pumps were shaking the ship in every member.
     “Thirty seconds!”
     “Plugs away! Ship power!” came LeRoy’s high-pitched voice. “Whoa! Port generator just quit!”
     “Switch to emergency!”
     “Emergency batteries on! The inverter’s getting hot!”
     “Fifteen seconds!”
     Chubb felt the skipper should call a hold as he said this last. If the inverter went out, the radar would lose its source of pulsed power, and a ship without radar was blind. But Louis said nothing.
     “Autopilot in command! Seven—six—five—four—”
     His voice was drowned out by a snarling, thundering, rippling, beating universe of noise.

     Chubb never knew there could be so much noise.
     It shook the bulkheads and rattled the deck plates. It bounced Chubb up and down on the couch pads. The mighty thrust of the Absyrtis’ rocket engines hammered at the structure of the ship.
     “Ship is away!” somebody screamed.
     There was a sudden, backsnapping jolt and Chubb knew that the breakaways on the guy lines holding the ship had failed. The lines had merely parted, but two-centimeter steel cable does not give way easily.
     He sneaked a quick glance at the tv monitor, but all he saw was a malestrom of sand, flaming gases, and the litter and sheds of Luna Louis’ junk yard being scattered all over the desert.
     Then the ship really began to shake as the combustion vibration of the rocket engines reinforced and excited the natural resonance of the hull. It jarred Chubb’s teeth even though he Was being compressed into his pads by the force of five gravities of acceleration. Instrument needles were bouncing wildly, so he quit looking at them; he couldn’t see them anyway because he was being shaken so hard.
     He lost all sense of time. After seeming hours, he felt the vibration build up to the point where he had to shut his eyes and hold on with all he had. The increased vibration told him that the ship was passing sonic speed, and that in turn would bring blessed relief from the flooding noise.
     Then there was no sound except the rattle and shake of the ship’s old plates and the thunderous whine of the pumps in the tail. He could hear the scream of the dyna- motors in the electronics compartment aft and the Whistling note of the doppler radar as is climbed up—and up—and up the musical scale until it Was an ear-splitting screech. All of this he heard through a gray haze. He couldn’t breathe; he was pinned to his couch, his heart racing and his anns flattened against their rests. And he knew why high body mass was a disqualifying factor in space flight. His body muscles were no stronger than a lighter man’s, yet they had to support more apparent mass under acceleration.
     The take-offs of the antipodal rockets had been nothing like this! He could feel this in his face, in his bones, in his entrails. With the noise and the acceleration, he felt nearer death than he had ever been. If this doesn’t stop, he thought wildly and dismally, if it doesn’t stop, I’ll die! I can’t stand this much longer! I can’t stand it!
     Whang! Clank! Slam!
     “Cut-off” LeRoy’s voice screamed breathlessly over the interphone.
     There were three more rough jolts, and the force holding him to his couch suddenly disappeared completely. His stomach made a violent attempt to eject itself through his throat, and he convulsed involuntarily. Then his eyes came into focus and a sensation akin to dizziness overcame him. The control room was suddenly over on its side, then upside-down, then right-side-up again. Then his astrostat hood swam upwards in front of him, the panel following it. It began to spin to the right, then stopped and sank toward him.
     But something has gone wrong! was his thought. He still felt weight! They couldn’t be in free-fall!
     Chubb had had limited experience in sub-gravity on the antipodal rocket trips; it was not entirely new to him. It was diiferent, but not new. Before his autonomic nervous system could build up a “storm”, he got a hold of himself. It took a moment for the nystagmus of his eyes to stop. Then he remembered the injection. With practiced movement, he tried to bring his hand to the pocket in the arm rest—and over-shot the mark by a foot. On the second try, he got his fingers firmly around the ampule and gave himself his shot right through his coveralls into his thigh muscles.
     It helped. He gripped the arm rests and shouted into the interphone, “Al! divisions report!”
     “Power room here!” LeRoy’s voice came back.
     Good old LeRoy! Maybe his heart isn't as bad as everybody thought!
     “Power plant in cut-off!” LeRoy Went on. “Pumps running down! Tanks holding pressure! But the reactor heat-exchanger is running too hot! We can’t get it down!”
     “Louis told me that might be normal! Keep your pumps running if you have to! How’s that generator?”
     “Out like a light! But the inverter’s holding! We’re okay. But, buddy, that was rough!”
     “Shipmaster reporting,” Greg called in. “My baliwick’s running. I’ve got a lot of sick passengers and I’m not feeling too chipper myself. Doc Barcarez is giving drop-shots or knock-outs as the case requires.”
     “Electronics Report!” Chubb snapped after a few seconds’ silence. “Bert! Report!”
     Bert’s head appeared instead in the aft hatch. His face was pale and drawn, his dark hair and horn-rimmed glasses giving him a ghostly appearance. There was a cut under his left eye with blood streaming in all directions over his face. “I’ve got a busted intercom,” he reported, “but one of my boys has a busted jaw. Thanks for the billy!”
     Chubb nodded and gulped, moving his hand slowly over to press against his stomach. He salivated freely for a second, then decided he wasn’t going to lose everything after all.
     “Bert Eggstrom will relinquish the command of his division to his senior engineer and report to the control room to assume the duties of the executive officer. Power officer, secure your power room from lift and proceed with underway activities. Electronics, notify White Sands Traffic Control of the situation and have them stand by for my formal report later. We’ll also need some radar fixes very shortly.” He switched off the interphone and turned to Bert. “Set up Program Number Two on the computer and get ready to process data. Move, while I get this star fix!” He pulled the astrostat hood down over his eyes while Bert seated himself before the computer console by the chart desk and began to program the computer.
     He scribbled the star angles on a pad attached to the astrostat and started making a preliminary determination. Actually, it was no star fix, but a sight on Venus, Jupiter, Luna, and Sol for Euler angles of ship attitude. On the short jump to Luna, he would rely heavily on the precision of radar and doppler data. The trajectory was too short and too simple for him to bother with stellar methods.
     He finished his calculations to find Bert hovering near him. “Does is check?” the new first mate asked.
     “Looks good. I think we can live with it.”
     “Computer’s ready. Bob Danforth's taken over below for me and has the radar data ready.”
     When they all showed up-some of them looking half-dead—Chubb was still running the computer. He checked the answer and satisfied himself with it, then set the device up to run a continuous program of trajectory by presenting x-, y-, and z-plots on three chart recorders.

From CONTRABAND ROCKET by Lee Correy (G. Harry Stine) (1956)

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

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