THE ROCKET EQUATION AND SMALL ROCKETS

Ed LeBouthillier

The Rocket Equation is one of the most basic tools used by orbital launch vehicle designers to help design their rockets. It was originally publicly published by Konstantin Tsiolkovsky. With minor additional considerations, it provides a powerful tool for launch vehicle designers. In this article, I will discuss the rocket equation and how it is used in rocket design.

HISTORY

The Rocket Equation became best known when it was published by Konstantin Tsiolkovsky, a Russian scientist, in 1903. It has been shown, however, that the equation was known to British military designers before it was published by Tsiolkovsky. But the equation is widely attributed to him.

Tsiolkovsky was a prolific writer very much interested in air and space travel. This interest led him to explore the mathematics of numerous methods of space propulsion and transportation. As well as the Rocket Equation, Tsiolkovsky also first mentioned the possibilities of space elevators.

DELTA V BASICS

In order to get into Earth orbit, a launch vehicle must accelerate to orbital velocity, which is 7.8 km/s (or about 25,500 feet per second or about 17400 miles per hour). As it sits on the launch pad, the launch vehicle's velocity is 0 km/s and therefore must "delta V" or change its velocity to 7.8 km/s. This is the meaning of "delta V;" it represents the amount that the velocity has to change.

Generally, an orbital launcher has to make this velocity change from zero to 7.8 km/s in a fairly short time, on the order of less than 6 minutes, so the average acceleration the rocket must generate from lift-off is about 21 m/s per second (over 2 g's). Often, rockets accelerate at higher rates. Think about this yourself. If you must accelerate to a velocity of 7800 m/s from a standing stop in about 6 minutes or less, you have to accelerate pretty significantly to do it.

THE ROCKET EQUATION IN DETAIL

The Rocket Equation describes the amount of velocity change (delta V) that a rocket experiences if no other external forces are experienced by it other than the thrust of the rocket motor. This means that it is in a vacuum and there are no gravitional influences affecting it. Although idealized in this form, it still represents an important set of relationships that guide rocket designers.

The basic formula is:

        dV = Ve * ln( Mf / Me )

Where:

        dV = delta Velocity or change in speed

        Ve = exhaust velocity of gas from rocket motor

        Mf = Fueled Mass of the rocket

        Me = Empty Mass of the rocket

This equation says that dV (the delta Velocity or change in velocity) is equal to Ve, the velocity with which mass is expelled, times the natural logarithm of the mass of the vehicle before the mass expulsion divided by the mass of the vehicle after the mass expulsion.

The full mass of the vehicle is generally seen as being composed of three major subcomponents: the propellant mass, the structure mass and the payload mass:

        Mf = Mp + Ms + Ml

Where:

        Mf = Full Mass of the rocket

        Mp = Mass of the propellant

        Ms = Mass of the rocket structure (tanks and motor)

        Ml = Mass of the payload

Additionally, the empty mass of the vehicle, Me, is composed of the following subcomponents:

        Me = Ms + Ml

Where:

        Me = Emtpy mass of the rocket (after expulsion of mass)

        Ms = Mass of the rocket structure (tanks and motor)

        Ml = Mass of the payload

A BIT MORE ON EXHAUST VELOCITY

It is worth considering another relationship that makes the rocket equation more useful. A relationship between the exhaust velocity of a rocket motor and its fuel efficiency has been established. This relationship is expressed as:

        Ve = g * Isp

Where:

        Ve  = Rocket exhaust velocity

        g   = gravitational acceleration (9.8 m/s^2 or 32.2 ft/sec^2)

        Isp = Specific Impulse

This equation says that the exhaust velocity from a rocket is related to its Specific Impulse (represented as Isp) times gravity.

SPECIFIC IMPULSE

Specific Impulse describes the efficiency of a rocket in producing thrust with a particular propellent combination. The equation for Specific Impulse is:

                Thrust * Seconds

        Isp =   ----------------

                 Propellant Mass

If the units are shown:

                       Kilograms-force * seconds

        (Seconds) =    -------------------------

                             kilograms

An alternate ways of looking at the units are:

                       Pounds-force * seconds

        seconds =      ----------------------

                              Pounds

It can be interpreted as being the number of seconds that 1 unit mass of propellant will burn when it produces one unit force of thrust. Since the forces in the numerator are in units of force expressed as weight (kg or lbs) times seconds and the propellant mass in the denominator is also in units of weight (kg or lbs), the weight units cancel to produce a final units of seconds.

This is a very important relationship for rocket designers. Different propellants have different Specific Impulses and knowing this relationship can help determine the amount of propellant needed.

Typical values of Isp are shown in the table below:

Propellant Combination	Isp
Black Powder	80 seconds
Potassium Nitrate &Sugar	138 seconds
Ammonium Perchlorate & HTPB	236 seconds
Nitric Acid & Kerosene	268 seconds
Liquid Oxygen & Gasoline	242 seconds
Liquid Oxygen & Kerosene	300 seconds
Liquid Oxygen & Hydrogen	440 seconds

A BIT ON MASS RATIO

The value of Mf/Me is known as the rocket equation's Mass Ratio. It specifies the ratio between a full rocket vehicle and the empty rocket vehicle. As said earlier:

        Mf = Ms + Mp + Ml

        Me = Ms + Ml

Again, where:

        Mf = Fueled mass of the rocket with payload

        Me = Emtpy mass of the rocket with payload

        Mp = Mass of the propellant

        Ms = Mass of the rocket structure (tanks and motor)

        Ml = Mass of the payload

Therefore:

        Mf/Me = (Ms + Mp + Ml) / (Ms + Ml)

If we extract this ratio from the rocket equation, we get:

        Mf/Me = 2.71828 ^ (dv/(g * Isp))

This ratio represents a relationship between the performance of the rocket vehicle and its mass. The table below shows the relationship between the Isp, the delta V and the mass ratio.

As the graph shows, in order to get a specific delta V, you can do it either by having a low Isp with a higher mass ratio, or a high Isp with a lower mass ratio. There are an infinite combinations of Isp and Mass Ratio which can give you a specified delta V. This is one of the major tasks of rocket design: choosing the Isp and Mass Ratio trade-off.

Since the Mass Ratio is defined by the ratio of three different system components: Mp, Ms and Ml, once you've fixed the delta V and the Isp, and usually your goal is to get a specified payload mass (Ml) into orbit, the critical factors in engineering a rocket is to control the two other mass components: the mass of the rocket (Ms) and the amount of propellant it carries (Mp) while also ensuring you get the specified Isp.

ENHANCING THE ROCKET EQUATION

With our knowledge of the relationship between Isp and Exhaust Velocity, we can make a simple substitution in the rocket equation which makes it even more useful. Knowing:

        Ve = g * Isp

We can substitute this relationship into the rocket equation to get this form:

        dV = Isp * g * ln( Mf / Me )

This form, although slightly more complicated, allows us to relate the delta Velocity, the rocket motor efficiency (in the form of Isp) with the mass before and after expulsion.

ADAPTATION TO ATMOSPHERIC FLIGHT

Although the rocket equation describes rocket flight without gravity or atmosphere, it has been adapted for more general use by the inclusion of factors which correct for its deficiencies. In particular, by accounting for the losses of gravity, air drag and other losses, it can be a valuable general purpose tool. In this case, the equation takes a format similar to:

        dV = g * Isp * ln( Mf / Me ) - Lg - La - Ls

where:

               dV  = delta Velocity

               g   = acceleration of gravity

               Isp = Specific Impulse of propellant

               Mf  = Fueled mass of rocket

               Me  = empty mass of rocket

               Lg  = losses due to gravity

               La  = losses due to aerodynamic drag

               Ls  = losses due to propellant used for steering

This augmented rocket equation accounts for velocity losses due to gravity, air drag and propellant used to steer the vehicle.

By subtracting out estimates of these other losses from the ideal form of the rocket equation, a useful estimate is derived of the delta Velocity for when the launcher is affected by gravity and air drag.

GRAVITY LOSSES

A rocket in a gravitational field experiences an acceleration that, when the rocket is flying away from the source mass, slows it down. This amount of slowdown is known as "Gravity Losses." If the rocket were pointing straight up while accelerating from Earth, gravity would pull it downward and slow it down. In the simplest case, the equation for defining the gravity loss is:

        Lg = g * t

Where:

               Lg = Gravity Loss of velocity

               g  = gravitational acceleration

               t  = duration of rocket thrust

Therefore, if we know that a rocket motor is going to burn for a time, t, we can get a good estimate of the gravity loss. This form of the equation only works for vertical flight, but there are forms which account for flight in a non-vertical trajectory.

It should be noted that this equation also only applies to the time that the rocket motor is thrusting as it represents the loss in delta V experienced for the burn duration.

So, if a rocket is thrusting vertically for 10 seconds, then the gravity loss during that time is:

        Lg = 9.80665 m/s^2 * 10 s

        Lg = 98.0665 m/s

One significant thing to realize is that higher thrust, and thus vehicular acceleration, results in a shorter flight time. Therefore, to minimize gravity losses, a higher acceleration is desired.

However, higher acceleration in an atmosphere can increase aerodynamic losses. So, an optimization between gravity losses and aerodynamic losses must be made for maximum overall delta V.

Typical gravity losses are dependent upon the flight path and time in flight, however, they can be seen as being in the ballpark of 1000 m/s.

AERODYNAMIC LOSSES

When a rocket is flying through the atmosphere, it will also experience a drag force which will tend to slow it down: air drag. Over the duration of the rocket motor burn, this drag accumulates to produce a loss in speed. This value, the Aerodynamic Loss, is difficult to determine using analytic methods (e.g. solving equations). Normally, it must be determined by using air tunnels, analogous flights of similar rockets, or numerical techniques. It is also possible to make order of magnitude estimates from aerodynamic losses of other vehicles.

In order to use numerical techniques, a simulated rocket is flown through a simulated atmosphere over the desired trajectory and the total aerodynamic losses are determined by integrating the instantaneous drag over the flight.

However, it is possible to know a good "first order estimate" of what the aerodynamic loss is from other examples. The following table shows some aerodynamic losses from previous rockets.

Vehicle	Mass (kg)	Aerodynamic Losses (m/s)
Apollo Saturn Launch Vehicle	3,039,000	40
Falcon 1	47200	93
Iranian Safir 2 LV	26000	80
Chubby Small Launch Vehicle	550	400
Slender Small Launch Vehicle	550	100

The important thing to note here is that as the rocket mass decreases, the aerodynamic losses increase. Additionally, one can decrease the aerodynamic losses by increasing the "slenderness" of the launch vehicle (by making it longer in height and narrower in diameter).

SCALING AERODYNAMIC LOSSES

A general scaling law relating the mass of the vehicle to the amount of aerodynamic losses is:

               (Mo / Mn)

        SFa =   ---------

               (Do/Dn)^2

Where:

        SFa = Aerodynamic Scaling Factor

        Mo  = Original Mass

        Mn  = New Mass

        Do  = Original Diameter

        Dn  = New Diameter

Taking the examples above, if the Saturn V launch vehicle was 10 meters in diameter and weighed 3039000 Kg, and we scaled it down to the size of a Falcon 1, then we have:

               (3000000 / 47200)      64.39

        SFa =   --------------- =      ------ = 1.79

               (10 / 1.67)^2          35.85

We would expect the Falcon 1 to have about 1.8 times the aerodynamic drag of the Saturn V or 40 m/s * 1.79 = 71.6 m/s aerodynamic losses. This is pretty close to the numerically derived value of 93 m/s aerodynamic losses. Now, it should be realized that this scaling estimate is only a rough one. Therefore, take the results from the scaling law skeptically and use numerical methods. However, one can at least get an order of magnitude answer from the scaling law.

The following table shows the predicted aerodynamic losses using this scaling law.

Vehicle	Aerodynamic Losses (m/s)	SFa Predicted Loss (m/s)
Apollo Saturn Launch Vehicle	40	40
Falcon 1	93	72
Iranian Safir 2 LV	80	73
Chubby Small Launch Vehicle	400	371
Slender Small Launch Vehicle	100	88

Obviously there is some error in the prediction using this formula, but it gives an order of magnitude estimate which seems usable. One reason for the errors is that there are other factors than scaling that effect aerodynamic losses (such as coefficient of drag). However, by multiplying by a safety factor, one should be able to get pretty accurate estimates. One reasonable safety factor to use is 1.14. By multiplying the SFa value by this, we get a bit closer to actually observed values.

PUTTING IT ALL TOGETHER

Let's take an example from what we’ve learned so far. Let's say that we want to make a rocket able to go to 121,000 feet. Let’s presume this basic vehicle is 8 inches in diameter, and 14 feet long, and will use a solid rocket motor. This composite solid rocket motor is to burn its complete thrust in 8 seconds with an Isp of 210 seconds. The rocket has a payload of 10 lbs, and the rest of the rocket weighs 160 lbs (not counting propellant). We need to determine how much propellant is needed and the final weight of the rocket.

Here’s a basic sketch:

Based on our earlier determination of the raw delta V required, we get:

S = ½ * a * t^2

121000 = ½ * 32.2 * t^2

t = 86.7 seconds

We can then figure out the unadjusted delta v as:

V = a * t

V = 32.2 * 86.7

V = 2791.7 fps

Knowing the motor burn time, we can figure out the gravity losses as:

V = a * t

V = 32.2 * 8

V = 257.6 fps

Finally, we need to determine the aerodynamic losses. We will use the Apollo Saturn as the baseline:

        (Mo / Mn)

SFa =   ---------

        (Do/Dn)^2

SFa = (3,039,000/145.1) / (10 / 0.20)^2

SFa = 20944 / 2500

SFa = 8.4

La = 1.14 * SFa * 40 m/s

La = 1.14 * 8.4 * 40 m/s

La = 1.14 * 336 m/s

La = 1.14 * 1102 fps

La = 1256 fps

So, the total delta V is:

dV = altitude_dv + La + Lg

dv = 2791.7 fps + 257.6 fps + 1256 fps

dv = 4305.3 fps

We can now start setting up the rocket equation:

Dv = g * Isp * ln(Mf/Me)

4305.3 = 32.2 * 210 seconds * ln( Mf/170 )

4305.3 = 32.2 * 210 * ln( Mf/170 )

4305.3/(32.2 * 210) = ln(Mf/170)

0.63669 = ln(Mf/170)

Mf = 170 * e ^ 0.63669

Mf = 170 * 2.71828 ^ 0.63669

Mf = 170 * 1.890

Mf = 321 lbs

Therefore, the parameters of the rocket are:

Isp = 210 seconds

Me = 170 lbs

Mf = 321 lbs

Ms = 160 lbs

Mp = 151 lbs

Ml = 10 lbs

A Sugar Rocket Example

Let’s presume that we want to make a sugar rocket able to go up to 121,000 feet. The Isp of sugar propellants is lower than composite propellants; they are able to provide about 130 seconds of Isp. We’ll use the payload and rocket weight from the last example.

In the last example, we already figured out the vacuum delta V as being 2791.7 fps and the target delta V as being 4305.3 fps. We’ll use these as first-order approximations to get closer to a final design.

Putting the information from the last example into the rocket equation gives us:

Dv = g * Isp * ln(Mf/Me)

4305.3 = 32.2 * 130 *ln(Mf/Me)

4305.3/(32.2*130) = ln(Mf/Me)

1.0285 = ln(Mf/Me)

e^1.0285 = Mf/Me

2.71828^1.0285=Mf/Me

2.80 = Mf/Me

If we break down Mf and Me into each of its components, we get:

2.80 = (Ml+Ms+Mp)/(Ms+Ml)

And then we remember that we have zero payload (Ml=0), then:

2.80 = (Ms+Mp)/Ms

2.80Ms=Ms+Mp

2.80Ms-Ms=Mp

Mp=1.80Ms

So, if we use the same structural weight from the previous example, Ms = 170, then the propellant needs to weigh

1.8 * 170lbs = 306 lbs

Giving us the following:

Isp = 130 seconds

Ms = 170 lbs

Mp=306 lbs

Ml=10 lbs

Me=170 lbs

Mf=476 lbs

The sugar rocket requires almost exactly double the propellant weight compared to the first example. This is largely due to the fact that the Isp is about half of that of the first example.

Summary

We’ve looked at the rocket equation and some of its implications. We’ve seen how  mass aspects affect the performance of the rocket. And we’ve seen how different propellant performances affect the weight of the rocket.