Structural Mass Fraction

In spacecraft design, when talking about the structural mass fractions, the NASA website gives the following equation:

Source: https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/mass-ratios/ However some sources divide the structural mass by the total mass (i.e. propellant mass + structural mass + payload mass).

My question is, when should the payload be included and when should it not be included in the calculation of structural mass fractions? Thanks.

• What's one of the other sources that use the other definition? Also note that epsilon is defined as the structural coefficient not mass fraction. Commented Apr 20 at 12:20
• Hi I have added a link (in the original post above) to one of the sources that mention the use of total mass in the denominator. I have also seen it being done this way in some lectures.
– G11
Commented Apr 20 at 15:43
• I'm with Marble, it looks like you've taken two similar but different things and lumped them both into "mass fractions". Use the structural coefficient when it's called for, and use the mass fraction when it's called for. The paragraph right after the one you quoted for the first source has an explanation explicitly stating $\epsilon$ is meant to be a design parameter independent of paylaod mass. Commented Apr 20 at 18:21
• Ok thank you both for your replies!
– G11
Commented Apr 20 at 21:31

Intro

The answer to your question is context-dependent. The context being what kind of engineering problems are you trying to solve and what are you trying to optimize. Whether is it more important to be as accurate as possible or better to arrive at an answer quickly (i.e. making a rule of thumb) may also factor in. It might help to think of a term like "structural mass fraction" or "structural coefficient" as a tool in your engineering toolbox. Like a Phillips screwdriver, there can be more than one kind even though they all serve essentially the same purpose. With experience, you will learn which screwdriver to reach for or whether you can get away with using the wrong screwdriver if you left the right screwdriver in your truck.

But allow me to describe a situation where accuracy is more important than speed and where the "right screwdriver" would be the "structural mass fraction" which includes the payload in the denominator.

An Educational Scenario

Lots of people are familiar with the rocket equation, but this equation doesn't tell you the mass of the payload as a function of parameters such as delta-v and $$v_{exhaust}$$, which is sometimes the information that you are primarily interested in obtaining.

What it does give you is the so-called "final mass", $$m_f$$. To get the payload mass, you need to subtract away the portion of the final mass that is "not payload", such as the rocket's dry mass and any propellants held in reserve for a boost-back burn, landing burn, and margin of safety.

To arrive at the payload mass, you need to determine the mass of the non-payload part of $$m_f$$. A portion of "non-payload" scales roughly in proportion to the initial mass of the rocket. This includes propellant tanks, engines, plumbing, etc. Another portion of non-payload is independent of the initial mass of the rocket. This might include, for example, a guidance computer. There are probably a few components that fall somewhere in between.

We can now introduce the concept of a "structural mass fraction" as an attempt to characterize the mass of the non-payload part of $$m_f$$ as a fraction of a rocket equation input parameter, such as the initial mass of the rocket.

For a rocket to achieve its mission (e.g. to reach orbit with a certain payload) it must be engineered to have a sufficiently low structural mass fraction (however we define it) and sufficiently high exhaust velocity. The baseline rocket equation doesn't include any kind of structural mass fraction term though. If we can create an exponential function like the rocket equation that does include a structural mass fraction term, then we could chart a more useful relationship between delta-v and $$m_0/m_{payload}$$. This chart will represent the so-called "tyranny of the rocket equation" more accurately.

To illustrate this, take a look at the chart below for a hypothetical rocket. The orange curve is the $$m_0/m_f$$ ratio that the bear-bones rocket equation will give you. With more knowledge (i.e. the structural mass fraction) you can derive an equation that will generate the blue curve, which is $$m_0/(m_{payload}+m_{avionics})$$, where $$m_{avionics}$$ is defined to mean the parts of the rocket, such as a guidance computer, that do not scale with $$m_0$$. With an even more sophisticated analysis, you can include staging which will allow you to generate the green curve. As the analysis becomes more sophisticated, the curves will become more accurate depictions of the rocket's true payload capabilities as a function of delta-v.

The Right Choice for This Scenario

My question is, when should the payload be included and when should it not be included in the calculation of structural mass fractions?

Assuming that your goal is to learn how to chart more accurate performance curves for rockets, your question poses two possible ways to introduce the concept of a "structural mass fraction".

In the first, which is more like the "structural coefficient" in the beginners guide to aeronautics, $$m_{payload}$$ is subtracted from $$m_0$$ in the denominator

$$k1={{m_{final}-m_{payload}} \over {m_0-m_{payload}}}={{"structure"} \over {m_0-m_{payload}}}={{m_s}\over{m_{propellant}+m_s}}$$

In the second, which is more like the "structural mass fraction" concept, $$m_0$$ is left alone

$$k2={{m_{final}-m_{payload}} \over {m_0}}={{"structure"} \over {m_0}}$$

In the beginner's guide article, the author characterizes the structural coefficient (k1) as follows

This parameter is independent of the payload that is launches

Aside from the grammatical error, I don't think this statement is true in this scenario. That article defines the structural coefficient more like k1, and k1 includes $$m_{payload}$$ in in denominator, which makes the structural coefficient less independent of $$m_{payload}$$.

Put another way, k2 is more "payload independent" because if you are an engineer tasked with reducing k2, you cannot report to your bosses that you made progress by reducing $$m_{payload}$$ (not that you would ever do that). Whereas you could attempt such a ruse with k1 - if your bosses were not paying close attention. The point is, that if you want to compare two similar rockets, if do this with k2 it will be less necessary to ask clarifying questions about the payload.

For example, let's suppose there were multiple versions of Starship and we were interested in plotting how the structural coefficient (k1) or structural mass fraction (k2) of the booster improved over time. If SpaceX kept changing the spit between the mass of the booster and the mass of its payload (that is, the Starship orbiter, its propellant, and its payload) from variant to variant, then the points on our Superheavy booster progress chart would be more affected by that if we used the structural coefficient (k1) and less affected by that if we used the structural mass fraction (k2).

Therefore, in the scenario described above, I would recommend starting with the "structural mass fraction" (k2) because it is defined to use $$m_0$$ in its entirety and does not require the payload to be subtracted out of the denominator.

That said, there might be another scenario where the best choice might be different.

• Phil, whilst this is interesting, it doesn't answer the OP's question. Please provide an answer to their question, or delete this post. Commented Apr 21 at 15:59
• Concur with Rory. Your not being able to think of a reason to use the structural coefficient is only evidence that this isn't an answer to OP's "when should it be included and when should it not." $\epsilon$ is defined as a payload-independent booster parameter, per OP's original source. Think of a booster that lofts to a lot of different target orbits--why not have one number that characterizes it, instead of one that alters for every target orbit because each supports a different maximum payload? Commented Apr 21 at 19:24
• Your edit where you claim that $\epsilon$ is actually "less independent of $m_payload$ [sic]" is twisted logic. It definitionally doesn't include a payload mass term. It cannot depend on payload. It's not a "ruse." Calling it one is harmful to the understanding that OP seeks. Commented Apr 21 at 20:58
• I never called ϵ "a ruse". Your last comment twists my words. It misrepresents what I said, and then argues that my post is "harmful" because it contains a statement that was never made (other than in the comment itself). BTW, am I correct in assuming that it was you who voted to delete? Commented Apr 21 at 21:46
• @Puffin Thanks for the feedback! I think it helped me improve the post a lot. See the changes above. Commented Apr 22 at 17:09