What is the story behind specific impulse being expressed in seconds?

Question

I've heard a couple of explanations but none of them quite make sense. One was that the German rocket engineers used metric and the American rocket engineers used the English system and seconds were the only thing they could agree on. Is that true?

Answer 1

It's even simpler than a German-American disagreement. It's use of ambiguous units.

The term "specific X" means the amount of X you can get from a unit mass of something. For instance, in batteries, specific energy means the total amount of energy you can get from one unit mass of battery. As described in the Wikipedia article, *specific impulse" is the total impulse you can get from a unit mass of propellants. Total impulse is just force times the duration of the force (time), which is equal to the momentum imparted to the object the force is applied to.

But in the early days in the US, propellant quantities were measured on scales calibrated in pounds, or more accurately, pounds-force: the force resulting from one Earth g acting on one pound-mass. The two are not equivalent! The Wikipedia article about English and US customary unit systems describes the consistent resolution of these units — but unfortunately these were not used in the early rocket labs, like Goddard's.

When you do it right, as in proper SI units, total impulse is in Newton-seconds, and mass is in kilograms. Dividing the former by the latter gives meters per second, a velocity: the rocket's exhaust velocity.

But when you use the "pound" as both mass and force and treat them as equivalent, total impulse is pound-seconds and mass is pounds. When you divide the two the "pounds" cancel and you're left with just seconds.

It turns out that this number with units of seconds indeed has a physical interpretation: If you burn one pound-mass of propellant in an engine that produces one pound-force of thrust, that is the duration, in seconds, the engine can operate before exhausting the one pound-mass propellant reservoir.

To get the exhaust velocity from that version of specific impulse you have to multiply by g, the true conversion constant between pounds-force and pounds-mass.

Answer 2

The term "specific" means "per unit." Specific impulse is defined as the impulse available per unit of fuel, and it has units of seconds when weight is chosen as the dividing unit. See this derivation by NASA for a step-by-step derivation of $I_{sp}$. I'll cover the important steps in this answer without relying on a unit system.

We begin with the equation for impulse: $$I = m V_{eq}$$

Dividing by weight: $$ \frac{I}{W} = \frac{m V_{eq}}{W} = I_{sp}$$

By Newton's second law, $F = ma \Rightarrow W = mg$

Substituting $mg$ for $W$: $$I_{sp} = \frac{m V_{eq}}{mg} = \frac{V_{eq}}{g}$$

As per the NASA derivation, $V_{eq} = \frac{F}{\dot{m}}$. Making this substitution: $$I_{sp} = \frac{\frac{F}{\dot{m}}}{g} = \frac{F}{\dot{m} g}$$

Now, let's do dimensional analysis: $$\left[I_{sp}\right] = \frac{\left[force\right]}{\frac{\left[mass\right]}{\left[time\right]} \frac{\left[length\right]}{\left[time\right]^2}}$$

Again by Newton's second law, we have $\left[force\right] = \left[mass\right] \frac{\left[length\right]}{\left[time\right]^2}$

The right-hand-side expression is found in the denominator of the previous equation. We have, finally, $$\left[I_{sp}\right] = \frac{\left[force\right]}{\frac{\left[force\right]}{\left[time\right]}} = \left[time\right]$$

So, both in US Customary units and SI units, $I_{sp}$ has units of seconds when weight is used as the dividing unit. It has nothing to do with agreeing upon units to use or "doing it right" and using "proper units."

Answer 3

The specific impulse $I_{\mathrm{sp}}$ in time units has the following interpretation. By burning propellant equal to a small fraction $\alpha$ of the rocket's total mass, the rocket can hover stationary in the air (in Earth gravity) for a short time $\tau$. The specific impulse is the proportionality factor: $\tau = \alpha I_{\mathrm{sp}}$. By integration, the rocket's total mass decreases with elapsed time $t$ as $\exp(-t/I_{\mathrm{sp}})$. That is, $I_{\mathrm{sp}}$ is the time constant for the decay of the rocket's mass when hovering.

And, since accelerating the rocket upward can only burn propellant faster, $I_{\mathrm{sp}}$ in time units sets a rough upper bound on how long a rocket can be fired during a launch from Earth. For example, the Space Shuttle's solid rocket boosters had a specific impulse of ~4 minutes and were fired for ~2 minutes.