# What about Torchships?

## Introduction

The term "Torchship" was coined by Robert Heinlein in 1953 for his short story "Sky Lift", it is also featured in his stories Farmer in the Sky, Time for the Stars, and Double Star. Sometimes it is referred to as "Ortega's Torch".

Nowadays it is implied that a Torchship is some kind of high thrust fusion drive, but Heinlein meant it to mean a handwavium total-conversion mass-into-energy type drive.

## What is a Torchship?

What is the definition of a torchship? Well, it is kind of vague. It more or less boils down to "unreasonably powerful."

With most propulsion systems, there is an inverse relationship between thrust and specific-impulse/exhaust-velocity (if one is high the other is low). So Rick Robinson defines a torch drive as propulsion system with both high acceleration (from high thrust) *and* high exhaust velocity. Note that whether the drive is "high acceleration" or not depends upon the thrust of the drive and the total mass of the **spacecraft**, that is, it does not just depend upon the drive.

If you recall, a propulsion systems **Thrust Power** (F_{p}) is thrust times exhaust velocity, then divide by two. Which sort of combines Rick's two parameters defining a torch drive (but so does propellant mass flow). The Drive Table is helpfully sorted in order of increasing Thrust Power. Remember that each entry is for a single engine of that type, it is possible to have multiple engines. With multiple engines, the exhaust velocity stays the same, but the thrust is multiplied by the number of engines.

A spacecraft's **Specific Power** (F_{sp}) is its propulsion system's Thrust Power divided by the spacecraft's dry mass. Remember that dry mass is the mass of spacecraft fully loaded with cargo and everything, but no propellant.

Rick Robinson's rule of thumb is that **a Torch Drive is a propulsion system with both both high acceleration and high thrust.** His further rule of thumb is that **a Torchship is a spacecraft with a Torch Drive and a specific power of one megawatt per kilogram or larger**.

And as a side note, such a frightful amount of thrust power in your spacecraft exhaust could do severe damange to anything it hosed. One would almost say *weapons-grade* levels of damage. Clever readers have already mouthed the phrase **the Kzinti Lesson**. This might lead to torchships being reserved for miliary vessels only.

Another definition of a torchship is **a spacecraft with more than 300 km/s total delta V and an acceleration greater than 0.01 g**. Which may or may not fit with Rick's definition.

I am trying to triangulate on Rick's rule of thumb for Torch Drives. I'm toying with defining "high thrust" as 100,000 Newtons or higher, and "high exhaust velocity" as 100,000 m/s or higher.

And a thrust power of 100 gigawatts or higher. Which is a number I pulled out of the air by examining the drive table and picking a dividing line that pleased me.

And a propellant mass flow between 100 and 0.01 kg/sec.

I'm working on it, OK? Let me crunch some numbers and draw a few graphs and I'll get back to you.

Would Rick Robinson consider the good ship *Polaris* to be a torchship? Let's see.

Our design equips the *Polaris* with not one, not two, but **three** freaking Nuclear Salt Water rocket engines. 12,900,000 Newtons of thrust each for a total of 38,700,000 Newtons. And an exhaust velocity of 66,000 meters per second. So NSWRs are definitely torch drives. Is the *Polaris* a torchship?

- F
_{p}= ( (F * E) * V_{e}) / 2 - F
_{p}= ( (12,900,000 * 3) * 66,000 ) / 2 - F
_{p}= ( 38,700,000 * 66,000 ) / 2 - F
_{p}= 2,554,200,000,000 / 2 - F
_{p}= 1,280,000,000,000 watts or 1.28 terawatts

The *Polaris* has a dry mass of 126,000 kilograms (126 metric tons).

- F
_{sp}= F_{p}/ M_{e} - F
_{sp}= 1,280,000,000,000 / 126,000 - F
_{sp}= 10,200,000 watts per kilogram or 10.2 megawatts per kilogram

Oh, yes, Rick Robinson would say the *Polaris* is very much a torchship.

## Torchship Performance

Fast Torchship High Gear Low Gear Wet Mass 1,000,000 kg Dry Mass 500,000 kg Mass Ratio 2 Exhaust

Velocity300,000 m/s 50,000 m/s ΔV 200,000 m/s 40,000 m/s Thrust 3,000,000 N 14,700,000 N Starting

Acceleration3 m/s ^{2}

(0.3 g)14.7 m/s ^{2}

(1.5 g)Thrust Power 450 GW 370 GW Let us say that our ship has a mass of 1000 tons, and a modest exhaust velocity of 300 km/s, a mere 0.001 c. In the name of further modesty we will set our acceleration at less than one third g, 3 meters per second squared. This ship is very much on the low end of torchships; Father Heinlein would hardly recognize her. If our torch has VASIMR style capability, we can trade specific impulse for thrust; dial exhaust velocity down to 50 km/s to develop 1.5 g, enough to lift from Earth. (but see below)

(ed note: trading specific impulse for thrust is shifting from high gear to low gear)Setting surface lift aside, let's look at travel performance from low Earth orbit. We will reach escape velocity about 15 minutes after lighting up, a good Oberth boot, but with this ship it hardly matters.

(ed note: 15 minutes of 1.5 g is 900 seconds of 14.7 m/s^{2}which is a ΔV cost of about13,000 m/s)Suppose half our departure mass is propellant, giving us about 180 km/s of delta v in the tanks

(ed note: 200,000 m/s total - 13,000 m/s liftoff =.187,000 m/s)Since we must make departure and arrival burns, our transfer speed (relative to Earth) is about 90 km/s.

(ed note: 180,000 / 2 =.90,000 m/s)Holding acceleration to 0.3 g as we burn off mass, we'll reach 90 km/s in less than eight hours,

(ed note: 90 km/s divided by 0.3 g is 90,000 m/s / 3 m/s.^{2}= 30,000 seconds. 30,000 / 3,600 =8.3 hours)...at about three times lunar distance from Earth. (If we were going to the Moon, we'd need to swing around three hours out for our deceleration burn.)

For any deep space mission we'll end up coasting most of the way, and 90 km/s is some pretty fast coasting. Even this low end torchship lets you take retrograde hyperboloid orbits, which is Isaac Newton's way of saying you can pretty much point & scoot.

You will truck along at 1 AU every three weeks, reaching Mars in little more than a week at opposition, though up to two months if you travel off season. The near side of the asteroid belt is a month from Earth, Jupiter three months in season. For Saturn and beyond things do get prolonged.

Still, 90 days for the inner system out to Jupiter is nothing shabby, and there's no apparent High Magic involved.

How much power output does out modest torch drive have? The answer, skipping simple but tedious math, is not quite half a terawatt, 450 GW.

(ed note: (3,000,000 N thrust * 300,000 m/s exhaust velocity) / 2 = 450,000,000,000 watts =.450 GW)This baby has 600 million horses under the hood. That's effective thrust power. Hotel power for the hab is extra, and so is getting rid of the waste heat.

Putting it another way, this drive puts out a tenth of a kiloton per second, plus losses.

(ed note: 450,000,000,000 watts / 4.184 × 10.^{12}=0.108 kilotons per second)And yes, Orion is the one currently semi plausible drive that can deliver this level of power and performance. I say semi plausible because, broadly political issues aside, I suspect that Orion enthusiasts gloss over the engineering details of deliberately nuking yourself thousands of times. Like banging your head on the table, it only feels good when you stop.

Reasonably Fast Torchship Wet Mass 1,000,000 kg Dry Mass 500,000 kg Mass Ratio 2 Exhaust

Velocity300,000 m/s ΔV 200,000 m/s Thrust 290,000 N Starting

Acceleration0.29 m/s ^{2}

(30 mg)Thrust Power 45 GW Really Fast Torchship Wet Mass 1,000,000 kg Dry Mass 500,000 kg Mass Ratio 2 Exhaust

Velocity3,000,000 m/s ΔV 2,000,000 m/s Thrust 3,000,000 N Starting

Acceleration3 m/s ^{2}

(0.3 g)Thrust Power 4.5 TW The good news, for practical, Reasonably Fast space travel, is that we don't need those near-terawatt burns. They don't really save much time on interplanetary missions — we could reduce drive power by 90 percent, and acceleration to 30 milligees — the acceleration of a freight train — and only add three days to travel time.

(Our reduced 'sub-torch' fusion drive is still putting out a non trivial 45 gigawatts of thrust power, which happens to be close to the effective thrust power of the Saturn V first stage.)

(ed note: (290,000 N thrust * 300,000 m/s exhaust velocity) / 2 = 45,000,000,000 watts =.45 GW)If you want Really Fast space travel, however, you will need more than this. Go back to our torchship and increase her drive exhaust velocity by tenfold, to 3000 km/s, and mission delta v to 1800 km/s, while keeping the same comfortable 0.3 g acceleration.

A classic brachistochrone orbit, under power using our full delta v, takes a week and carries us 270 million km, 1.8 AU.

- transitDeltaV = 2 * sqrt[ D * A ]
- ((transitDeltaV / 2)^2) / A = D
- ((1,800,000 / 2)^2) / 3 = D
- 270,000,000,000 m = D
- 270,000,000 km = D
- 270 million km = D
Add a week of coasting in the middle and you're at Jupiter. Saturn is about three weeks' travel, and you can reach distant Eris in 6 months.

Drive power output of our upgraded torchship is now 4.5 TW, about a third the current power output of the human race. Which in itself is no argument against it. Controlling the reaction and getting rid of the waste heat are more immediate concerns.

(ed note: (3,000,000 N thrust * 3,000,000 m/s exhaust velocity) / 2 = 4,500,000,000,000 watts =.4.5 TW)As for a true, Heinleinian torchship? Heinlein's torch is a mass conversion torch. He sensibly avoids any details of the physics, but apparently the backwash is a mix of radiation, AKA photon drive, and neutrons, probably relativistic.

Torchship

Lewis and Clark, pictured above, is about 60 meters in diameter, masses in on the order of 50,000 tons, and in Time for the Stars she begins her relativistic interstellar mission by launching from the Pacific Ocean at 3 g. I don't know how to adjust the rocket equations for relativity, but the naive, relativity-ignoring calculation gives a power output of 225 petawatts, AKA 225,000 TW, AKA 53 megatons per second.Do not try this trick at your homeworld. I don't know whether Heinlein never checked this calculation, did it and ignored the results, or did it and decided that a few dozen gigatons — 12 torchships are launched, one after another — was no big deal since it was off in the middle of the Pacific somewhere.

Maybe he only did the calculation later, because in other stories the torchships sensibly remain in orbit (served by NERVA style nuke thermal shuttles).

## Torch Drive Heat

*Magnetic nozzle from Discovery II*

*Painting by Vincent Di Fate for the novel***Starfire**by Paul Preuss. Note magnetic nozzle. Click for larger image

The main problem with torch drives is getting rid of the waste heat generated by such a monster.

Remember that **Thrust Power** (F_{p}) is thrust times exhaust velocity, then divide by two. Thrust power is rated in watts, and if you look at the ratings for torch drives, well, that's a lotta watts.

Unavoidably some of those watts are going to become waste heat.

If you have a Resistojet with a thrust power of 700 watts, and 10% becomes waste heat, the heat will be 70 watts. This is only about as much as emitted by an old style incandescent bulb, not a problem.

But if you have Rick's Fast Torchship with a thrust power of 450 gigawatts, the heat will be 45,000 megawatts. Since your average drive assembly can survive no more than 5 megawatts, it will glow blue-white for a fraction of a second before it vaporizes. Along with most of the spacecraft.

A spacecraft with a chemical engine can have a torchship like specific power of 1MW/kg, but the chemical engine is not a torch drive because of its pathetic exhaust velocity. Chemical engines have no waste heat problem. Yes they have huge amounts of it, but they can easily use their huge propellant mass flow to get rid of it. Basically the exhaust plume acts like a heat radiator. The technical term is "**open-cycle cooling**".

Unfortunately torch drives cannot use that trick. High propellant flow equals pathetic specific impulse. Torch dives have large specific impulse ratings, which means low propellant flows, which means they will have to rely upon something else to keep themselves from melting. There isn't enough propellant in the exhaust plume to carry away the heat.

**mDot = F / V _{e}**

where:

- F = Thrust
*(Newtons)* - mDot = Propellant Mass Flow
*(kg/s)* - Ve = Exhaust Velocity
*(m/s)*

As you can see from the above equation, if both thrust and specific impulse is torche-drive high, the propellant mass flow will be small.

So open-cycle cooling is out. If your drive has an enclosed reaction chamber made of matter, Anthony Jackson's rule of thumb is:

**R _{c} = 0.12 * sqrt[H]**

where

- R
_{c}= reaction chamber radius*(meters)* - H = reaction chamber waste heat
*(megawatts)*

As a first approximation, for most propulsion systems one can get away with using the thrust power for **H**. Science-fictional technologies can cut the value of **H** to a percentage of thrust power by somehow preventing the waste heat from getting to the chamber walls.

Only use this equation if **H** is above **4,000 MW** or so, and if the propulsion system is a **thermal** type *( i.e., fission, fusion, or antimatter)*.

Say your propulsion system has an exhaust velocity of 5.4 × 10

^{6}m/s and a thrust of 2.5 × 10^{6}N.Now

Fso the thrust power is 6.7 × 10_{p}=(F*V_{e})/2^{12}W. Divided by 1.0 × 10^{6}watts per megawatt gives us6.7 × 10.^{6}megawattsPlugging this into the equation results in 0.12 * sqrt[6.7e6 MW] = drive chamber radius of

310 metersor a diameter of a third of a mile.Ouch.

What about Rick's Fast Torship with 450 gigawats? That is 450,000 megawatts. Plugging this into the equation results in a chamber radius of only 81 meters, diameter of only about the length of a US football field. Still freaking ginormous.

So enclosing a thermal torch drive inside a reaction chamber made of matter appears to be a dead end.

Therefore, the main strategy is to try and direct the drive energy with magnetic fields instead of metal walls.

Rick Robinson says that even subtorch high-end drives will have an open latticework to support a magnetic containment nozzle. He likes to call it a "lantern", because it will glow brilliantly.

Configuration of a shield designed to protect a superconducting coil against neutron radiation. The tungsten surface of the shield is inclined at an angle of 200-1 to the source of radiation, so that the neutrons will glance off.

Artwork by Luke CampbellAnd as for torch ships — a bigger problem than creating the hellish inferno of nuclear fire needed to propel your spacecraft is keeping your spacecraft from evaporating under the intense x-ray, gamma-ray, and neutron irradiation.

My best guess for accomplishing this is keeping the torch flame outside the spacecraft, and couple the plasma to the spacecraft using magnetic fields to give you thrust. The latter trick probably means several conductive — or, more likely, superconductive — loops of cable surrounding the torch flame but at a good healthy radius.

To protect your field generating cable, you will need to shield them, probably with blade-like tungsten structures edge-on to the torch. The large surface area of the blades gives you a lot of radiator surface while only intercepting a small amount of radiation (only what is needed to shield the cables), tungsten does a good job of scattering neutrons away without heating up much, at shallow angles of incidence x-rays (but not neutrons or gammas) are reflected away, and tungsten can heat up to yellow-white hot without evaporating too much.

An unusual consequence of the latter, and the relatively small emissions from your optically thin thrust plasma, is that visually the tungsten shields will be the brightest part of your spacecraft when under thrust — the intensity will be the same as that of an incandescent filament from a light bulb, but you will have a lot more area to radiate from.

These resulting torch craft don't end up looking much like conventional rockets. You get loopy filigree and from the support and field cables, and graceful glowing sheets from the heat shields, enclosing a volume that is probably much larger than the passenger/payload section.

*Fusion drive spacecraft from Attack Vector: Tactical. A Luke Campbell set of blade shields would probably be several times the diameter of these. Artwork by Charles Oines.*

Click for larger image

Artwork by Steve BowersTorch drive describes a class of thrusters for spacecraft propulsion in which a high energy yield detonation or pulse is initiated at a high rate external to the spacecraft. Magnetic fields are used to deflect the plasma produced by the pulse to generate thrust. By detonating the pulse outside the main structure of the spacecraft, the neutral radiation by-products of the pulse (neutrons, bremmstrahlung x-rays, gamma rays, and thermal radiation) can mostly escape into space, without creating a large thermal load aboard the spacecraft. This allows very high energy thrusters, which can combine both high thrust and high delta-V simultaneously.

A torch drive requires one or more current carrying loops surrounding the reaction region to produce the magnetic field that deflects the plasma and charged radiation from the drive pulses. Typically, these loops are made of superconductors, since any normal conductor would quickly melt or vaporize under the high currents needed to produce the field. The field coils are backed by a high tensile strength support to withstand the magnet-current back reaction from bursting the superconductor.

These field coils must be protected from the intense neutral radiation that the drive pulses produce. This is because a superconductor that becomes too hot will cease to superconduct. A common shielding design is a sheet of tungsten with a "V"-shaped cross section at a very narrow opening angle, resembling a knife-edge. The point of the V faces the radiation source, the field coil runs along the open top of the V. At very small angles of incidence, tungsten makes a good reflector of x-rays so that most x-rays are simply reflected at low angles away from the field coils and into space. Since tungsten atoms are much heavier than neutrons, a collision between a tungsten nucleus and a neutron results in the neutron rebounding with most of its original energy, delivering only 1% of its energy on average to the tungsten shield. This scatters the neutrons away from the field coils. The narrow opening angle means that the tungsten knife edge is essentially a sheet perpendicular to the incoming radiation, allowing a large radiator area compared to the cross section exposed to the radiation.

Rates of tungsten sublimation become problematic at temperatures above 3000 K, so the shield is typically placed far enough away from the drive pulses to keep its temperature at or below this value. At all times, the shield must be kept below 3695 K, the melting point of tungsten. At these temperatures, the tungsten knife-edge sheets are radiating at a blazing yellow-white color, with the intensity of an M-class star or an old style incandescent bulb filament. A torch drive in operation appears as a brilliant flare primarily from the thermal radiation of the tungsten shields — the drive pulses themselves emit relatively little visible light in comparison. With a 200:1 aspect ratio for the length of the knife blade to its width, a heat shield that absorbs 1% of the incident radiation and scatters the rest can withstand an incident intensity of 90 GW/m

^{2}.A plasma in a magnetic field will expand against the field while the energy density of the plasma is greater than the energy density of the field. In SI units, the energy density of the field in J/m

^{3}is given by

B^{2}/(2*mu_0)where

Bis the magnitude of the magnetic field in tesla andmu_0is the magnetic constant (mu_0 = 4 pi * 10^{-7}N/A^{2}). If the field coils produce a uniform field, then a drive pulse with energyEin its plasma will expand against the field until it reaches a volumeVsuch that the following relation approximately holds

V = 2 mu_0 E/B^{2}Assuming the radius is approximately spherical, the pulse's blast will expand into a fireball with a diameter

dapproximately the cube root of this volume. The timetit will take for the pulse to expand to a stop before it is deflected is approximately

t = d/V_exwhere

V_exis the velocity of the torch drive exhaust. If a second pulse is detonated before the first pulse has been fully deflected, it will add its energy to that of the first and require a larger volume to hold the combined fireball.The size of the field coils is set by the requirement that the tungsten shield remain cool enough not to sublimate, and for the drive pulse fireballs not to contact the tungsten shield. Since the tungsten shield extends a considerable distance toward the detonation point (the distance from the field coil to the tip of the shield is typically 200 times or more the width of the field coil), the field coil must be set far enough back that the tip of the tungsten shield does not evaporate.

If there are multiple field coils producing the magnetic field, then the knife-blade heat shield of one coil will scatter neutrons and radiate thermal heat onto the heat shields of the other field coils, compromising their ability to shed heat. Consequently, many designs use only a single field coil despite the loss of efficiency. Those torch drives that use multiple field coils typically space them at distances significantly larger than the length of the knife-blade shield.

As previously mentioned, the tungsten knife-blade heat shield that protects the field coils will glow very brightly. At 3000 K, it will radiate 4.6 MW/m

^{2}of heat as thermal radiation. As an example, consider a torch drive with 2 cm wide field coils, 4 meter long knife-blade heat shields that are exposed to 90 GW/m^{2}, and 10 MW of neutral radiation produced by the drive pulses. At this rated intensity, the tip of the heat shield can be as close as 9 meters to the detonation point. This produces a radiating disk with an outer radius of 13 meters and an inner radius of 9 meters, which will therefore radiate 2.6 GW of heat and 76 Glm of luminous power combined from its front and back. The apparent brightness will depend on the angle of the disk with respect to the observer, but unless the disk is edge-on, the unaided dark-adapted eye of a baseline human could detect the disk at a distance of approximately 1 Gm (gigameter) and would appear as an apparent magnitude 0 point of light at approximately 0.05 Gm. For comparison, the distance between Sol and old Earth is approximately 150 Gm. A dedicated 1 meter aperture early alarm scanning scope could detect the disk at approximately 600 Tm (terameter) with 1 kilosecond exposure time. For comparison, this is over 1% of the distance from Sol to its nearest stellar neighbor, Alpha Centauri.

## Brachistochrone Equations

*Artwork by Tim Early (1996)*

As RocketCat pointed out, torchships are unobtanium. Therefore, it doesn't mean the math no longer applies. You just need different equations.

First figure the distance between the two planets, say Mars and Terra. The *"superior"* planet is the one farthest from the Sun, and the *"inferior"* planet is nearest. The distance from the Sun and the superior planet is D_{s} and the distance between the Sun and the inferior is D_{i}. No "church lady" jokes please.

Obviously the maximum distance between the planets is when they are on the opposite sides of the Sun,
the distance being D_{s} + D_{i}. And of course the minimum is when they are on the same side, distance being D_{s} - D_{i}. Upon reflection you will discover that the average distance between the planets is D_{s}. *(when averaging, D _{i} cancels out.)*

Just choose a distance between the max and min. If you want to actually calculate the distance between two planets on a given date, be my guest but I'm not qualified to explain how. Do a web search for a software "orrery".

A Hohmann orbit is the maximum transit time / minimum deltaV mission. Weak spacecraft use this because they do not have a lot of deltaV.

A *"Brachistochrone"* is a minimum transit time / maximum deltaV mission. Torchships use this because they have lots of deltaV to spare.

You accelerate constantly to the midpoint, flip over *("skew flip")*, and decelerate to the destination. Weaker torchships will accelerate up to a certain velocity, coast for a while, then decelerate to rest. Brachistochrone missions are not only of shorter mission time, but they also are not constrained by launch windows the way Hohmann are. You can launch any time you like.

*Playing counters from***Triplanetary**by GDW, 1972. Note torchship counter in upper right corner, the one with the propellant rating of "infinity".

It is very important to note that it takes *exactly* the same amount of time to slow from a speed X to speed zero as it took to accelerate from speed zero to speed X. There is no way to jam on the brakes for a stop-on-a-dime halt. Other than lithobraking.

People who played the ancient boardgame
**Triplanetary** or the new game Voidstriker discovered this the hard way. They would spend five turns accelerating to a blinding speed, find out to their horror that it would take five turns to slow down to a stop, and end up either streaking off the edge of the map or smacking into Mars fast enough to make a crater.

This is why a Brachistochrone accelerates to the **mid-way** point then decelerates the rest of the trip. The idea is to come to a complete stop at your destination.

*Torchship Lewis & Clark. Artwork by Bruce Lewis.*

If you know the desired acceleration of your spacecraft *(generally one g or 9.81 m/s^{2})* and wish to calculate the transit time, the Brachistochrone equation is

**T = 2 * sqrt[ D/A ]**

where

- T = transit time
*(seconds)* - D = distance
*(meters)* - A = acceleration
*(m/s*^{2}) - sqrt[x] = square root of x

Remember that

- AU * 1.49e11 = meters
- 1 g of acceleration = 9.81 m/s
^{2} - one-tenth g of acceleration = 0.981 m/s
^{2} - one one-hundredth g of acceleration = 0.0981 m/s
^{2}

Divide time in seconds by

- 3600 for hours
- 86400 for days
- 2592000 for (30 day) months
- 31536000 for years

Timothy Charters worked out the following equation. It is the above transit time equation for weaker spacecraft that have to coast during the midpoint

**T = ((D - (A * t^2)) / (A * t)) + (2*t)**

where

- T = transit time
*(seconds)* - D = distance
*(meters)* - A = acceleration
*(m/s*^{2}) - t = duration of acceleration phase
*(seconds)*, just the acceleration phase only, NOT the acceleration+deceleration phase.

Note that the coast duration time is of course = T - (2*t)

If you know the desired transit time and wish to calculate the required acceleration, the equation is

**A = (4 * D) / T ^{2}**

Keep in mind that prolonged periods of acceleration a greater than one g is very bad for the crew's health.

*"Colony Sphere" from a 1959 poster. Everything else in the poster has been "borrowed" from other sources, so one of suspicious mind would think this might have been "inspired" by the colony torchship "Mayflower" in Heinlein's***FARMER IN THE SKY**.

Don't be confused. You might think that the Brachistochrone equation should be **T = sqrt[ 2 * D/A ]** instead of **T = 2 * sqrt[ D/A ]**, since your physics textbook states that **D = 0.5 * A * T^2**. The confusion is because the **D** in the physics book refers to the *mid-way* distance, not the *total* distance.

This changes the physics book equation from

**D = 0.5 * A * t^2**

to

**D * 0.5 = 0.5 * A * t^2**

Solving for **t** gives us **t = sqrt(D/A)** where **t** is the time to the mid-way distance. Since it takes an equal amount of time to slow down, the total trip time **T** is twice that or **T = 2 * sqrt( D/A )**. Which is the Brachistochrone equation given above.

## Delta-V

### Calculating

*Torchship Mayflower from "Satellite Scout" (Farmer in the Sky) by Robert Heinlein. Artwork by Chesley Bonestell.*

Now, just how brawny a rocket are we talking about? Take the distance and acceleration from above and plug it into the following equation:

**transitDeltaV = 2 * sqrt[ D * A ]**

where

- transitDeltaV = transit deltaV required
*(m/s)*

The rocket will also have to match orbital velocity with the target planet. In Hohmann orbits, this was included in the total.

**orbitalVelocity = sqrt[ (G * M) / R ]**

where

- orbitalVelocity = planet's orbital velocity
*(m/s)* - G = 0.00000000006673
*(Gravitational constant)* - M = mass of primary
*(kg)*, for the Sun:**1.989e30** - R = distance between planet and primary
*(meters)*(semi-major axis or orbital radius)

If you are talking about missions between planets in the solar system, the equation becomes

**orbitalVelocity = sqrt[1.33e20 / R ]**

Figure the orbital velocity of the start planet and destination planet, subtract the smaller from the larger, and the result is the matchOrbitDeltaV

**matchOrbitDeltaV = sqrt[1.33e20 / D _{i} ] - sqrt[1.33e20 / D_{s} ]**

If the rocket lifts off and/or lands, that takes deltaV as well.

**liftoffDeltaV = sqrt[ (G * P _{m}) / P_{r} ]**

where

- liftoffDeltaV = deltaV to lift off or land on a planet
*(m/s)* - G = 0.00000000006673
- Pm = planet's mass
*(kg)* - Pr = planet's radius
*(m)*

The total mission deltaV is therefore:

**totalDeltaV = sqrt(liftoffDeltaV ^{2} + transitDeltaV^{2}) + sqrt(matchOrbitDeltaV^{2} + landDeltaV^{2})**

Do a bit of calculation and you will see how such performance is outrageously beyond the capability of any drive system in the table I gave you.

If you want to cheat, you can look up some of the missions in Jon Roger's Mission Table.

### Nomograms

*Transit Time Nomogram**Find the required engine for the given ship mass and desired acceleration on the Transit Time Nomogram.**Find the delta-v for the given engine and mass ratio on the delta-v nomogram.**Find the required delta-v and travel time for a given distance using the Transit Time Nomogram.**Decrease the acceleration if the required delta-v is too great.*

For some ballpark estimates, you can use my handy-dandy Transit Time Nomogram. A nomogram is an obsolete mathematical calculation device related to a slide rule. It is a set of scales printed on a sheet of paper, and read with the help of a ruler or straight-edge. While obsolete, it does have some advantages when trying to visualize a range of solutions. Print out the nomogram, grab a ruler, and follow my example. You can also purchase an 11" x 17" poster of this nomogram at . Standard disclaimer: I constructed this nomogram but I am not a rocket scientist. There may be errors. Use at your own risk.

Let's say that our spacecraft is 1.5 ktons *(1.5 kilo-tons or 1500 metric tons)*. It has a single Gas-Core Nuclear Thermal Rocket engine *(NTR-GAS MAX)* and has a *(totally ridiculous)* mass ratio of 20. The equation for figuring a spacecraft's total DeltaV is **Δ _{v} = Ve * ln[R]**. On your pocket calculator, 98,000 * ln[20] = 98,000 * 2.9957 = 300,000 m/s =

**300 km/s**. Ideally this should be on the transit nomogram, but the blasted thing was getting crowded enough as it is. This calculation is on a separate nomogram found here.

The mission is to travel a distance of 0.4 AU *(about the distance between the Sun and the planet Mercury)*. Using a constant boost brachistochrone trajectory, how long will the ship take to travel that distance?

Examine the nomogram. On the Ship Mass scale, locate the 1.5 kton tick mark. On the Engine Type scale, locate the NTR-GAS MAX tick mark. Lay a straight-edge on the 1.5 kton and NTR-GAS MAX tick marks and examine where the edge crosses the Acceleration scale. Congratulations, you've just calculated the ship's maximum acceleration:2 meters per second per second *(m/s ^{2})*.

For your convenience, the acceleration scale is also labeled with the minimum lift off values for various planets.

So we know our ship has a maximum acceleration of 2 m/s^{2} and a maximum DeltaV of 300 km/s. As long as we stay under both of those limits we will be fine.

On the Acceleration scale, locate the 2 m/s^{2} tick mark. On the Destination Distance scale, locate the 0.4 AU tick mark. Lay a straight-edge on the two tick marks and examine where it intersects the Transit time scale. It says that the trip will take just a bit under four days.

But wait! Check where the edge crosses the Total DeltaV scale. Uh oh, it says almost 750 km/s, and our ship can only do 300 km/s before its propellant tanks run dry. Our ship cannot do this trajectory.

The key is to remember that 2 m/s^{2} is the ship's *maximum* acceleration, nothing is preventing us from throttling down the engines a bit to lower the DeltaV cost. This is where a nomogram is superior to a calculator, in that you can visualize a range of solutions.

Pivot the straight-edge on the 0.4 AU tick mark. Pivot it until it crosses the 300 km/s tick on the Total DeltaV scale. Now you can read the other mission values: 0.4 m/s^{2} acceleration and a trip time of a bit over a week. Since this mission has parameters that are under both the DeltaV and Acceleration limits of our ship, the ship can perform this mission *(we will assume that the ship has enough life-support to keep the crew alive for a week or so)*.

Of course, if you want to have some spare DeltaV left in your propellant tanks at the mission destination, you don't have to use it all just getting there. For instance, you can pivot around the 250 km/s DeltaV tick mark to find a good mission. You will arrive at the destination with 300 - 250 = **50 km/s** still in your tanks.

Which reminded me that I had not worked out how long it would take to get home on a one-gee boost, if it turned out that I could not arrange automatic piloting at eight gees. I was stymied on getting out of the cell, I hadn't even nibbled at what I would do if I did get out (correction: when I got out), but I could work ballistics.

I didn't need books. I've met people, even in this day and age, who can't tell a star from a planet and who think of astronomical distances simply as "big." They remind me of those primitives who have just four numbers: one, two, three, and "many." But any tenderfoot Scout knows the basic facts and a fellow bitten by the space bug (such as myself) usually knows a number of figures.

"Mother very thoughtfully made a jelly sandwich under no protest." Could you forget that after saying it a few times? Okay, lay it out so:

MotherMERCURY $.39 VeryVENUS $.72 ThoughtfullyTERRA $1.00 MadeMARS $1.50 AASTEROIDS (assorted prices, unimportant) JellyJUPITER $5.20 SandwichSATURN $9.50 UnderURANUS $19.00 NoNEPTUNE $30.00 ProtestPLUTO $39.50 The "prices" are distances from the Sun in astronomical units. An A.U. is the mean distance of Earth from Sun, 93,000,000 miles. It is easier to remember one figure that everybody knows and some little figures than it is to remember figures in millions and billions. I use dollar signs because a figure has more flavor if I think of it as money — which Dad considers deplorable. Some way you must remember them, or you don't know your own neighborhood.

Now we come to a joker. The list says that Pluto's distance is thirty-nine and a half times Earth's distance. But Pluto and Mercury have very eccentric orbits and Pluto's is a dilly; its distance varies almost two billion miles, more than the distance from the Sun to Uranus. Pluto creeps to the orbit of Neptune and a hair inside, then swings way out and stays there a couple of centuries — it makes only four round trips in a thousand years.

But I had seen that article about how Pluto was coming into its "summer." So I knew it was close to the orbit of Neptune now, and would be for the rest of my life-my life expectancy in Centerville; I didn't look like a preferred risk here. That gave an easy figure — 30 astronomical units.

Acceleration problems are simple s=1/2 at

^{2}; distance equals half the acceleration times the square of elapsed time. If astrogation were that simple any sophomore could pilot a rocket ship — the complications come from gravitational fields and the fact that everything moves fourteen directions at once. But I could disregard gravitational fields and planetary motions; at the speeds a wormface ship makes neither factor matters until you are very close. I wanted a rough answer.I missed my slipstick. Dad says that anyone who can't use a slide rule is a cultural illiterate and should not be allowed to vote. Mine is a beauty — a K&E 20" Log-log Duplex Decitrig. Dad surprised me with it after I mastered a ten-inch polyphase. We ate potato soup that week — but Dad says you should always budget luxuries first. I knew where it was. Home on my desk.

No matter. I had figures, formula, pencil and paper.

First a check problem. Fats had said "Pluto," "five days," and "eight gravities."

It's a two-piece problem; accelerate for half time (and half distance); do a skew-flip and decelerate the other half time (and distance). You can't use the whole distance in the equation, as "time" appears as a square — it's a parabolic. Was Pluto in opposition? Or quadrature? Or conjunction? Nobody looks at Pluto — so why remember where it is on the ecliptic? Oh, well, the average distance was 30 A.U.s — that would give a close-enough answer. Half that distance, in feet, is: 1/2 × 30 × 93,000,000 × 5280. Eight gravities is: 8 × 32.2 ft./sec./sec. — speed increases by 258 feet per second every second up to skew-flip and decreases just as fast thereafter.

So — 1/2 × 30 × 93,000,000 × 5280 = 1/2 × 8 × 32.2 x t

^{2}— and you wind up with the time for half the trip, in seconds. Double that for full trip. Divide by 3600 to get hours; divide by 24 and you have days. On a slide rule such a problem takes forty seconds, most of it to get your decimal point correct. It's as easy as computing sales tax.It took me at least an hour and almost as long to prove it, using a different sequence — and a third time, because the answers didn't match (I had forgotten to multiply by 5280, and had "miles" on one side and "feet" on the other — a no-good way to do arithmetic) — then a fourth time because my confidence was shaken. I tell you, the slide rule is the greatest invention since girls.

But I got a proved answer. Five and a half days. I was on Pluto.

(Ed note: I learned it asMy

Very

Educated

Mother

Just

Served

Us

Nine

Pumpkins.

In Slide Rule terminology: K&E is Keuffel & Esser, noted manufacturer of quality slide rules. 20 inches is twice the size and accuracy of a standard slide rule. Log-log means the rule possesses expanded logarithmic scales. Duplex means there are scales on both sides of the rule and the cursor is double sided. Decitrig means the rule possesses decimal trigometric scales.)

**Have Space Suit - Will Travel**by Robert A. Heinlein, 1958

Thanks to Charles Martin for this analysis:

Find the required delta-v.Use the required delta-v and an assumed mass ratio to find the exhaust velocity.In Heinlein's short story "Sky Lift", the torchship on an emergency run to Pluto colony does 3.5 g for nine days and 15 hours. 3.5 g is approximately 35 m/s

^{2}and 9d15h is 831,600 seconds. 35 m/s^{2}* 831,600 s =29,100,000 m/s total deltaV.Assume a mass ratio of 4. Most of Heinlein's ships had a mass ratio of 3, 4 is reasonable for an emergency trip.

V

_{e}= Δ_{v}/ ln[R] so 29,100,000 / 1.39 =21,000,000 m/s exhaust velocityor seven percent of the speed of light.A glance at the engine table show that this is way up there, second only to the maximum possible Antimatter Beam-Core propulsion, and twice the maximum of Inertial Confinement Fusion. If Heinlein's torchship can manage a V

_{e}of ten percent lightspeed it can get away with a mass ratio of 3.

## Acceleration Tanks

**Earthlight**by Sir Arthur C. Clarke, 1955. Nice picture, but it does violate the "rockets point down" principle.

If a torchship is going to accelerate at more than one g for longer than a few minutes, the crew is going to need special couches to lie in. Otherwise the g forces will cause severe injury or even kill.

In "Sky Lift" and Double Star, the crew spent the days of high thrust in acceleration couches that were like advanced waterbeds *(called "cider presses")*. In **The Mote in God's Eye** by Larry Niven and Jerry Pournelle, the captain's chair had a built-in "relief tube" *( i.e., a rudimentary urinal)* for use during prolonged periods of multi-g acceleration. There were also a few motorized acceleration couches used by damage control parties who had to move around during high gs. Such mobile couches also appeared in Joe Haldeman's

**The Forever War**.

He called Bury instead.

Bury was in the gee bath: a film of highly elastic mylar over liquid. Only his face and hands showed above the curved surface. His face looked old—it almost showed his true age.

"Yes, of course, I didn't mean personally. I only want access to information on our progress. At my age I dare not move from this rubber bathtub for the duration of our voyage. How long will we be under four gees?"

"One hundred and twenty-five hours. One twenty-four, now."

He called Sally's cabin.

She looked as if she hadn't slept in a week or smiled in years. Blaine said, "Hello, Sally. Sorry you came?"

"I told you I can take anything you can take," Sally said calmly. She gripped the arms of her chair and stood up. She let go and spread her arms to show how capable she was.

"Be careful," Blaine said, trying to keep his voice steady. "No sudden moves. Keep your knees straight. You can break your back just sitting down. Now stay erect, but reach behind you. Get both the chair arms in your hands before you try to bend at the waist—"

She didn't believe it was dangerous, not until she started to sit down. Then the muscles in her arms knotted, panic flared in her eyes, and she sat much too abruptly, as if

MacArthur'sgravity had sucked her down."Are you hurt?"

"No," she said. "Only my pride."

"Then you stay in that chair, damn your eyes! Do you see me standing up? You do not. And you won't!"

"All right." She turned her head from side to side. She was obviously dizzy from the jolt.

**The Mote in God's Eye**by Larry Niven and Jerry Pournelle

Model by Dan Thompson. Click for larger imageModel by Dan Thompson. Click for larger imageModel by Dan Thompson. Click for larger image“How high, sir?”

Berrio hesitated. “Three and one-half gravities.”

Three and a half g’s! That wasn’t a boost — that was a pullout. Joe heard the surgeon protest, “I’m sorry, sir, but three gravities is all I can approve.”

Berrio frowned. “Legally, it’s up to the captain. But three hundred lives depend on it.”

Kleuger said, “Doctor, let’s see that curve.” The surgeon slid a paper across the desk; Kleuger moved it so that Joe could see it. “Here’s the scoop, Appleby—”

A curve started high, dropped very slowly, made a sudden “knee” and dropped rapidly. The surgeon put his finger on the “knee.” “Here,” he said soberly, “is where the donors are suffering from loss of blood as much as the patients. After that it’s hopeless, without a new source of blood.”

“How did you get this curve?” Joe asked.

“It’s the empirical equation of Larkin’s disease applied to two hundred eighty-nine people.”

Appleby noted vertical lines each marked with an acceleration and a time. Far to the right was one marked: “1 g—18 days” That was the standard trip; it would arrive after the epidemic had burned out. Two gravities cut it to twelve days seventeen hours; even so, half the colony would be dead. Three g’s was better but still bad. He could see why the Commodore wanted them to risk three-and-a-half kicks; that line touched the “knee,” at nine days fifteen hours. That way they could save almost everybody, but, oh, brother!

The time advantage dropped off by inverse squares. Eighteen days required one gravity, so nine days took four, while four-and-a-half days required a fantastic sixteen gravities. But someone had drawn a line at “16 g—4.5 days.” “Hey! This plot must be for a robot-torch — that’s the ticket! Is there one available?”

Berrio said gently, “Yes. But what are its chances?”

Joe shut up. Even between the inner planets robots often went astray. In four-billion-odd miles the chance that one could hit close enough to be caught by radio control was slim. “We’ll try,” Berrio promised. “If it succeeds, I’ll call you at once.” He looked at Kleuger. “Captain, time is short. I must have your decision.”

Kleuger turned to the surgeon. “Doctor, why not another half gravity? I recall a report on a chimpanzee who was centrifuged at high g for an amazingly long time.”

“A chimpanzee is not a man.”

Joe blurted out, “How much did this chimp stand, Surgeon?”

“Three and a quarter gravities for twenty-seven days.”

“He did? What shape was he in when the test ended?”

“He wasn’t,” the doctor grunted.

The ship was built for high boost; controls were over the pilots’ tanks, where they could be fingered without lifting a hand. The flight surgeon and an assistant fitted Kleuger into one tank while two medical technicians arranged Joe in his. One of them asked, “Underwear smooth? No wrinkles?”

“I guess.”

“I’ll check.” He did so, then arranged fittings necessary to a man who must remain in one position for days. “The nipple left of your mouth is water; the two on your right are glucose and bouillon.”

“No solids?”

The surgeon turned in the air and answered, “You don’t need any, you won’t want any, and you mustn’t have any. And be careful in swallowing.”

“I’ve boosted before.”

“Sure, sure. But be careful.”

Each tank was like an oversized bathtub filled with a liquid denser than water. The top was covered by a rubbery sheet, gasketed at the edges; during boost each man would float with the sheet conforming to his body. The

Salamanderbeing still in free orbit, everything was weightless and the sheet now served to keep the fluid from floating out. The attendants centered Appleby against the sheet and fastened him with sticky tape, then placed his own acceleration collar, tailored to him, behind his head.The room had no ports and needed none. The area in front of Joe’s face was filled with screens, instruments, radar, and data displays; near his forehead was his eyepiece for the coelostat. A light blinked green as the passenger tube broke its anchors; Kleuger caught Joe’s eye in a mirror mounted opposite them. “Report, Mister.”

“Minus seven’ minutes oh four. Tracking. Torch warm and idle. Green for light-off.”

“Stand by while I check orientation.” Kleuger’s eyes disappeared into his coelostat eyepiece.

When the counter flashed the last thirty seconds he forgot his foregone leave. The lust to travel possessed him. To go, no matter where, anywhere go! He smiled as the torch lit off.

Then weight hit him.

At three and one-half gravities he weighed six hundred and thirty pounds. It felt as if a load of sand had landed on him, squeezing his chest, making him helpless, forcing his head against his collar. He strove to relax, to let the supporting liquid hold him together. It was all right to tighten up for a pullout, but for a long boost one must relax. He breathed shallowly and slowly; the air was pure oxygen, little lung action was needed. But he labored just to breathe. He could feel his heart struggling to pump blood grown heavy through squeezed vessels. This is awful! he admitted. I’m not sure I can take it. He had once had four g for nine minutes but he had forgotten how bad it was.

Joe then found that he had forgotten, while working, his unbearable weight. It felt worse than ever. His neck ached and he suspected that there was a wrinkle under his left calf. He wiggled in the tank to smooth it, but it made it worse.

He tried to rest — as if a man could when buried under sandbags.

His bones ached and the wrinkle became a nagging nuisance. The pain in his neck got worse; apparently he had wrenched it at light-off. He turned his head, but there were just two positions — bad and worse. Closing his eyes, he attempted to sleep. Ten minutes later he was wider awake than ever, his mind on three things, the lump in his neck, the irritation under his leg, and the squeezing weight.

Look, bud, he told himself, this is a long boost. Take it easy, or adrenalin exhaustion will get you. As the book says, “The ideal pilot is relaxed and unworried. Sanguine in temperament, he never borrows trouble.” Why, you chair-warming so-and-so! Were you at three and a half g’s when you wrote that twaddle?

The integrating accelerograph displayed elapsed time, velocity, and distance, in dead-reckoning for empty space. Under these windows were three more which showed the same by the precomputed tape controlling the torch; by comparing, Joe could tell how results matched predictions. The torch had been lit off for less than seven hours, speed was nearly two million miles per hour and they were over six million miles out. A third display corrected these figures for the Sun’s field, but Joe ignored this; near Earth’s orbit the Sun pulls only one two-thousandth of a gravity — a gnat’s whisker, allowed for in precomputation. Joe merely noted that tape and D.R. agreed; he wanted an outside check.

His ribs hurt, each breath carried the stab of pleurisy. His hands and feet felt “pins-and-needles” from scanty circulation. He wiggled them, which produced crawling sensations and wearied him. So he held still and watched the speed soar. It increased seventy-seven miles per hour every second, more than a quarter million miles per hour every hour. For once he envied rocketship pilots; they took forever to get anywhere but they got there in comfort.

Without the torch, men would never have ventured much past Mars. E = Mc

^{2}, mass is energy, and a pound of sand equals fifteen billion horsepower-hours. An atomic rocketship uses but a fraction of one percent of that energy, whereas the new torchers used better than eighty percent. The conversion chamber of a torch was a tiny sun; particles expelled from it approached the speed of light.“Oh, there’s one thing I don’t understand, uh, what I don’t understand is, uh, this: why do I have to go, uh, to the geriatrics clinic at Luna City? That’s for old people, uh? That’s what I’ve always understood — the way I understand it. Sir?”

The surgeon cut in, “I told you, Joe. They have the very best physiotherapy. We got special permission for you.”

Joe looked perplexed. “Is that right, sir? I feel funny, going to an old folks’, uh, hospital?”

“That’s right, son.”

Joe grinned sheepishly. “Okay, sir, uh, if you say so.”

They started to leave. “Doctor — stay a moment. Messenger, help Mr. Appleby.”

“Joe, can you make it?”

“Uh, sure! My legs are lots better — see?” He went out, leaning on the messenger.

Berrio said, “Doctor, tell me straight: will Joe get well?”

“No, sir.”

“Will he get better?’

“Some, perhaps. Lunar gravity makes it easy to get the most out of what a man has left.”

“But will his mind clear up?”

The doctor hesitated. “It’s this way, sir. Heavy acceleration is a speeded-up aging process. Tissues break down, capillaries rupture, the heart does many times its proper work. And there is hypoxia, from failure to deliver enough oxygen to the brain.”

The Commodore struck his desk an angry blow. The surgeon said gently, “Don’t take it so hard, sir.”

“Damn it, man — think of the way he was. Just a kid, all bounce and vinegar — now look at him! He’s an old man — senile.”

“Look at it this way,” urged the surgeon, “you expended one man, but you saved two hundred and seventy.”

*Sky Lift*by Robert Heinlein (1953)

*Torchship Lewis & Clark. Artwork by Jon Stopa.*

A hand grabbed my arm, towed me along a narrow passage and into a compartment. Against one bulkhead and flat to it were two bunks, or "cider presses," the bathtub-shaped, hydraulic, pressure-distribution tanks used for high acceleration in torchships. I had never seen one before but we had used quite convincing mock-ups in the space opus The Earth Raiders.

There was a stenciled sign on the bulkhead behind the bunks: WARNING!!! Do Not Take More than Three Gravities without a Gee Suit. By Order of— I rotated slowly out of range of vision before I could finish reading it and someone shoved me into one cider press. Dak and the other men were hurriedly strapping me against it when a horn somewhere near by broke into a horrid hooting. It continued for several seconds, then a voice replaced it: "Red warning! Two gravities! Three minutes! Red warning! Two gravities! Three minutes!" Then the hooting started again.

I looked at him and said wonderingly, "How do you manage to stand up?" Part of my mind, the professional part that works independentiy, was noting how he stood and filing it in a new drawer marked: "How a Man Stands under Two Gravities."

He grinned at me. "Nothing to it. I wear arch supports."

"Hmmmph!"

"You can stand up, if you want to. Ordinarily we discourage passengers from getting out of the boost tanks when we are torching at anything over one and a half gees - too much chance that some idiot will fall over his own feet and break a leg. But I once saw a really tough weight-lifter type climb out of the press and walk at five gravities - but he was never good for much afterwards. But two gees is okay - about like carrying another man piggyback."

She did not return. Instead the door was opened by a man who appeared to be inhabiting a giant kiddie stroller. "Howdy there, young fellow!" he boomed out. He was sixtyish, a bit too heavy, and bland; I did not have to see his diploma to be aware that his was a "bedside" manner.

"How do you do, sir?"

"Well enough. Better at lower acceleration." He glanced down at the contrivance he was strapped into. "How do you like my corset-on-wheels? Not stylish, perhaps, but it takes some of the strain off my heart.

At turnover we got that one-gravity rest that Dak had promised. We never were in free fall, not for an instant; instead of putting out the torch, which I gather they hate to do while under way, the ship described what Dak called a 180-degree skew turn. It leaves the ship on boost the whole time and is done rather qulckly, but it has an oddly disturbing effect on the sense of balance. The effect has a name something like Coriolanus. Coriolis?

All I know about spaceships is that the ones that operate from the surface of a planet are true rockets but the voyageurs call them "teakettles" because of the steam jet of water or hydrogen they boost with. They aren't considered real atomic-power ships even though the jet is heated by an atomic pile. The long-jump ships such as the Tom Paine, torchships that is, are (so they tell me) the real thing, making use of F equals MC squared, or is it M equals EC squared? You know — the thing Einstein invented.

Our Moon being an airless planet, a torchship can land on it. But the Tom Paine, being a torchship, was really intended to stay in space and be serviced only at space stations in orbit; she had to be landed in a cradle. I wish I had been awake to see it, for they say that catching an egg on a plate is easy by comparison. Dak was one of the half dozen pilots who could do it.

**DOUBLE STAR**by Robert Heinlein, 1956