by John Walker

“Rocket science” has often been used as a metaphor for something complicated, arcane, or difficult. Well, as one who has from time to time been called complicated, arcane, and difficult myself, I’m going to try to explain the essentials of it to you in this brief essay. There will be some math, but nothing you can’t do on a spreadsheet, pocket calculator, or slide rule.

First, let me distinguish rocket *science*, the physics which
underlies rocket propulsion, which is relatively simple, from rocket
*engineering*, which encompasses disciplines including propellant
chemistry, thermodynamics, structural and vibration analysis,
aerodynamics, guidance and control theory, fluid dynamics, and
metallurgy. Rocket engineering is genuinely difficult, and is made
even more challenging since most rockets are not reusable and hence,
when you push the launch button, you’re trying out everything in a
real flight for the first time.

At the level of the basic science, however, a rocket is pretty simple. You have a structure, usually tubular in shape, which holds some kind of propellant. In a chemical rocket, the propellant(s) burn, producing hot gases which issue from a nozzle at one end of the tube. The ejection of these gases creates, under Newton’s third law of motion, a reaction force upon the rocket, causing it to accelerate. How much acceleration do we get by burning all the propellant in a rocket? That’s given by the rocket equation, which Konstantin Tsiolkovsky applied in the late 19th century in his investigations of the feasibility of space flight. Don’t be afraid: it’s pretty straightforward once you get past the notation. Let’s look it in the face.

Δ*v* (pronounced delta-vee) is the change in velocity which will
result from burning all of the propellant in the rocket. This will
have units like metres/second (m/s). The velocity of the exhaust
which comes out of the rocket is *v*_{e} in the same units. The total
mass of the rocket, including propellant, at the time of launch is
*m*_{0}. The mass of the rocket (structure, payload, engine(s),
avionics, etc.) which remains after all the fuel is burned is *m*_{f}.
(This is sometimes called the “dry mass”, although that doesn’t
really make sense in the case of solid rockets.) You can specify *m*_{0}
and *m*_{f} in any mass units (kilograms, tonnes, etc.) as long as they
are the same: only the ratio matters. The
natural logarithm function
is ln. The key thing you need to know about the ln function is that
it grows very slowly as its argument increases. This, as we’ll see,
is what makes rocketry so difficult.

You’ll often hear the performance of rocket engines and propellants
specified in terms of
specific impulse,
or *I*_{sp}. For rockets (as
opposed to, say, air-breathing engines), this is simply another way
of expressing the exhaust velocity, obtained by dividing the exhaust
velocity by the gravitational acceleration at the Earth’s surface.
If you have exhaust velocity in metres per second, you obtain *I*_{sp}
in units of seconds by dividing by the acceleration of gravity of
9.8 m/s².

One advantage of using *I*_{sp} is that it’s the same regardless of
which unit you use for length.

Let’s put some numbers on this to see the magnitude of what we’re talking about. Here are the exhaust velocities produced by some commonly used rocket propellants in modern, high-performance engines.

LOX is
liquid oxygen;
LH_{2} is
liquid hydrogen.
Both are gases which
must be liquefied and stored at
cryogenic
temperatures. Liquid
hydrogen must be kept even colder than LOX and, because its density
is low, requires large fuel tanks.
RP-1
is essentially kerosene, and
requires no special handling beyond that of diesel fuel or home
heating oil.
Nitrogen tetroxide
(N_{2}O_{4}) and
hydrazine
(N_{2}H_{4}) are
liquids at room temperature and can be stored indefinitely, ready to
use. They are
hypergolic
with one another: they burst into
combustion spontaneously on contact, so when used in rocket engines,
no igniter is required; this makes them attractive for engines which
must restart multiple times. Unfortunately, both nitrogen tetroxide
and hydrazine are highly toxic and must be handled with extreme
caution. A wide variety of solid rocket propellants are used. They
have both fuel and oxidiser mixed together, and require only an
igniter to set them burning. Obviously, great care must be taken
that they aren’t ignited prematurely, and once lit, they burn to
completion whether you like it or not. The one I’ve cited here is
ammonium
perchlorate composite propellant
(APCP). Other widely used
solid propellants are
HTPB
and
PBAN
(follow the links for their jaw-breaking names).

All right, enough with the equations, numbers, and acronyms. Let’s
go fly a rocket and see how all this stuff actually works. We’ll
begin with a very simple rocket, the
GEM-60
solid rocket booster
optionally used by the
Delta IV
launch vehicles. While it is
normally used as a booster in the early part of a launch and then
jettisoned, here we’re going to launch it by itself, just to see
what happens. We will have no payload: just the rocket itself. Fully
packed with propellant, the GEM-60 weighs 33,638 kg (*m*_{0}). At
burn-out, just 91 seconds later, the final mass (*m*_{f}) is but 3,941
kg. Exhaust velocity (*v*_{e}) from the HTPB solid fuel is 2401 m/s, for
an *I*_{sp} of 245 s. Plugging these numbers into the rocket equation,
we find Δ*v* to be 5148 m/s. Now, the rocket equation is for an ideal
rocket, operating perfectly efficiently in a vacuum without the
effects of air resistance or gravity, but let’s just go with that
number for the moment. This is pretty fast; in fact, “faster than a
speeding bullet” doesn’t do justice to it. The
5.56×45 NATO round
fired by the M-16 rifle has a muzzle velocity of 940 m/s, so our
rocket will not just be faster than the bullet, but more than *five
times faster*. Pretty impressive, but what if we want to go to orbit?

*Forget about it!* Orbital velocity in
low Earth orbit
is around 7.8
km/s, but when you take into account the effects of gravity and air
resistance during the first part of the rocket’s flight through the
thick atmosphere, you’ll need around 9.5 km/s of Δ*v* to get to orbit
if you’re launching toward the East reasonably close to the equator,
more otherwise. (When launching to the East, you get to add the
Earth’s rotation to your Δ*v* for free.) Now, you might think, from
the simple rocket we’ve just worked out, “Well, we’re half way
there. How hard can it be?” And now we come back to that natural
logarithm function in the rocket equation; it’s a killer. With a
little algebra we can rewrite the equation as:

The quantity on the left is called the
mass ratio:
the ratio of the
mass of the rocket at launch to the mass at burnout. Now you can see
that this ratio increases *exponentially* as the ratio of Δ*v* to
exhaust velocity grows. The mass ratio of our simple rocket was
33638/3941=8.5. In other words, just 12% of the rocket’s weight at
liftoff remained at burnout: at launch the rocket was 88%
propellant. All right, let’s crank the numbers for an orbital
launch, which requires a Δ*v* of 9500 m/s. Now watch that exponential
kick in! We get a mass ratio of, not 8.5, but 52! To reach orbit,
our rocket would have to be 98% propellant on the launch pad. If we
wanted to put one tonne (1000 kg) in orbit (including the rocket in
the payload), we’d need the rocket to mass 52 tonnes on the launch
pad. Not only is this ridiculously inefficient, we simply lack the
materials to build structures this light compared to the weight they
support.

What if we used a more energetic fuel? In the
Golden Age of science
fiction, many stories featured the discovery of a super-powerful
rocket fuel which would open up the space frontier, and intrigue and
derring-do involving access to the fuel. Experience has taught that
unless you want to work with stuff so
hideously toxic, volatile,
expensive, and recalcitrant that nobody in their right mind would
consider it, liquid oxygen and liquid hydrogen are the best you can
do when it comes to chemical propellants. They are efficient,
non-toxic, reasonably inexpensive, and their combustion product is
just water vapour. Let’s try a LOX/LH_{2} rocket and see how far we
get. We’ll start with the first stage of the
Delta IV launcher,
which burns those propellants. This stage hasn’t ever been launched
by itself, so we’ll add 500 kg for a nose cone, guidance system, and
avionics with which it isn’t usually equipped. The stage, fully
fuelled, has a mass of 228,400 kg and empty, 26,400 kg. Its
energetic fuel delivers an exhaust velocity of 4057 m/s, or an *I*_{sp}
of 414. Recall that exhaust velocity is in the denominator of that
ratio in the exponential: this is an *enormous* improvement over the
solid fuel booster! Our mass fraction, 8.7, is about the same, but
the energetic fuel gives us a Δ*v* of 8678 m/s. Much better, but we’re
still short of the 9500 or so we need to get to orbit, and we’re
still not carrying any payload.

We’re already using the highest energy fuel that’s practical, and
burning it in a
modern, high-performance rocket engine. The rocket
structure is so light it allows 88% of the liftoff weight to be
fuel. What is to be done? If only we could launch our rocket at a
higher altitude, with a kick to get it started, rather than having
to make it crawl all the way up from the ground through the dense
lower atmosphere, losing Δ*v* to the drag of gravity! Well, that’s
exactly what we’ll do.

From the early days of rocketry, engineers who worked out the
consequences of the rocket equation realised it was extraordinarily
difficult to get to orbit from Earth’s surface. But, being
engineers, they immediately asked themselves, “Can we cheat?” It
turns out, you can. The trick is to use a large rocket to launch, as
its payload, a smaller rocket. When the big rocket burns out, the
smaller rocket is ignited, except now it’s not being launched from
Earth, but rather with a high altitude and velocity, where its
engine will operate more efficiently and have to provide less than
the total Δ*v* to get to orbit. This idea was demonstrated at the end
of the 1940s by the
RTV-G-4 Bumper
program, which used a German V-2
to launch an American WAC Corporal rocket, which ultimately reached
an altitude of 393 km. The Bumper launch pictured at the left was
the first rocket launch from Cape Canaveral.

Now we’re going to get real. Let’s put some serious payload in low
Earth orbit. We’ll start with the first stage of the Delta IV we
used in the last example, but use it to launch a second stage:
another rocket, which carries the actual payload. The properties and
performance of the first stage are identical to those above. The
second stage, which is called the
Delta Cryogenic Second Stage
(CSS), has a launch mass of 30,710 kg and a mass at burnout of 3490
kg. It is also fueled with LOX/LH_{2}, but since its engine only
operates in a vacuum, is more efficient, with an exhaust velocity of
4628 m/s, or an *I*_{sp} of 462 s. On top of the second stage, we’ll
stack a payload of 8000 kg (including a
fairing
we’ll dispose of
after getting out of the atmosphere, but we’re doing science, not
engineering here, so let’s neglect those details).

Three—two—one—zero—*ignition*—liftoff! The stack (first stage, second
stage, and payload) lumbers off the launch pad. The numbers for the
first stage are the same, but we must add the mass of the second
stage and payload, which are (30710+8000)=38710 kg. As expected,
this reduces Δ*v* for the first stage, to 5092 m/s. But the second
stage starts its work at this sweet spot, made even sweeter because
it’s now outside the sensible atmosphere and runs more efficiently.
To the mass of the second stage, we add the payload of 8000 kg. When
the second stage burns out, it has contributed a Δ*v* of 4451 m/sec,
for a total for the two stages of 9543 m/s. We’ve delivered our
payload to orbit.

Consider how difficult this was to do. The payload, 8 tonnes, is now
in orbit. The total mass of the launcher was 267 tonnes less than
ten minutes before. Only 3% of the mass of the launcher remains when
the payload separates from the second stage; all the rest is junk.
Even without considering the innumerable engineering details needed
to accomplish such a mission, getting to orbit is *hard*. If the Earth
were only slightly more massive than it is, it would be impossible
to get to orbit using chemical propellants. Perhaps when we
encounter other spacefaring species, most will hail from smaller
planets, around the more numerous red dwarf stars. They’d be of
slight stature, and have large eyes adapted to the dimmer light of
their stars. But I’m not
going there.

Rocket science can be fun, and it isn’t complicated. All you need to work out “what if” scenarios is a spreadsheet or pocket calculator, and abundant information available in the public domain on the Internet.

Here are the fundamentals of rocket propulsion explained in 1950 by that eminent rocket scientist, Prof. Woody Woodpecker, from Destination Moon.

This is a TEDxHouston talk by astronaut Donald Pettit about the tyranny of the rocket equation.

Scott Manley demonstrates rocket science with Kerbal Space Program and a spreadsheet.

August, 2016

*This document is in the public domain.*