# When would the density Isp product be an important performance metric of a propellant?

The quote

NASA is testing hydroxylammonium nitrate in space which is expected to perform 50% better than standard propellants and can be allowed to freeze (hydrazine must be kept liquid). LMP-103S has been tested gets about 30%.

used in Why would it be “less bad” for hydroxylammonium nitrate monopropellant to freeze than for hydrazine? and found in this answer to *Is the EU really banning “toxic propellants” in 2020? How is that going to work? * raised a question about how a monopropellant could be 50% better than hydrazine.

The reply comment links to GPIM AF-M315E Propulsion System which explains that the performance metric is the product of density and Isp.

Question: When would the density Isp product be an important performance metric of a propellant?

From the ECAPS page on LMP-103S performance...

The specific impulse is ≥ 6% higher and the propellant density is 24% higher. As a result, the satellite can either be fitted with a smaller tank, or the mission duration can be extended while retaining the same tank size.

Volume is important because you only have so much room inside a payload fairing. Propellant available for station keeping and maneuvering is a limit on the lifespan of a spacecraft.

• Thanks! This might also apply to smallsats of fixed size, e.g. ambitious 3U cubesats. – uhoh Aug 2 '19 at 23:17
• fyi I just realized that "HAN" is another name for hydroxylammonium nitrate, and so it seems I've questioned about its use in cubesats before! Is HAN-water propellant “green” enough to ever be used in a cubesat? – uhoh Aug 2 '19 at 23:21

When would the density Isp product be an important performance metric of a propellant?

It's always important.

Specific impulse is a measure of impulse provided per mass unit of combusted propellant, which is a great metric for fuel-efficency in a rocket. However, it doesn't take into account the mass of things other than fuel -- like fuel tankage, which has mass generally proportional to its volume.

Hydrogen, for example, looks vastly better than kerosene if you just look at specific impulse, but it is about 1/15 as dense, so some of the specific impulse advantage is lost due to requiring much more tankage volume.

Density-Isp-product isn't a mathematically "correct" metric for comparing propellants -- you'd need a spreadsheet and information about tankage structure, pressure, etc. to actually compute a correct metric -- but it could be useful for identifying interesting candidate propellants before moving onto the spreadsheet phase of the contest.

Oh, I see.

Let's say I am building an ambitious 3U cubesat and I've allocated 1U for fuel and 2U for payload and bus, and I want to select the propellant that will give me the most delta-v.

I don't know Cubesat mass density (kg/U) statistics? but lets say I've got a 3 kg (dry mass $$m_D$$) 3U cubesat with a 1 liter tank. What is the delta-v as a function of $$I_{sp}$$ and $$\rho$$ with a fixed tank volume $$V$$? Applying tyrannical Tsiolkovsky we get:

$$\Delta v = I_{sp} g_0 \ln \frac{m_D+\rho V}{m_D} = I_{sp} g_0 \ln \left(1 + \frac{\rho V}{m_D}\right).$$

Remember that the log of 1 plus something small is approximately that thing (i.e. $$\ln(1+x) \approx x$$ for small $$x$$)

$$\Delta v \approx I_{sp} g_0 \frac{\rho V}{m_D} = I_{sp}\rho \frac{g_0 V}{m_D}.$$

The density specific impulse product $$I_{sp}\rho$$ is the predictor of total delta-v possible when the tank volume is fixed, and the relationship is simple linear when the propellant mass is a small fraction of the total spacecraft mass.

For flying fuel tanks you'll need to keep the logarithm in the expression and so while close, the highest $$I_{sp}\rho$$ product monopropellant may not have exactly the highest $$\Delta v$$. checking now...

• The logarithm is a concave function, so for a fixed $I_{sp}\rho$ product (and the same dry mass and tank volume), scaling $I_{sp}$ up and $\rho$ down by the same factor will increase $\Delta v$. Which makes sense — you're still getting the same total impulse (i.e. change in momentum) from burning the fuel, but have less mass to push during the burn. – Ilmari Karonen Aug 5 '19 at 22:09