Not sure, if I should "answer" or "comment", but let's put some more questions...
If I have a two stage launch and I still want to stick up with the $9000 \text {km/s} \Delta V$, should I then first calculate the fractions of propellant I need for the two stages?
$$M_f = 1 - (M_1/m_0) = 1 - e^{-\Delta V/V_e}$$
And If the first stage would be until $5000 \text {m/s}$ and second until $9000 \text {m/s}$ with $V_e = 4500 \text {m/s}$, I would get:
$$M_{f1} = 1 - e^{-5000/4500} = 67.1% \text {propellant}$$
$$M_{f2} = 1 - e^{-4000/4500} = 58.9% \text {propellant}$$
With the second one, I need to use the deltaV from $5000$ to $9000 \text {m/s}$ ($=4000 \text {m/s}$), right?
With assumption of $8%$ from the mass being construction, I end up with
First stage:
- construction $8%$
- propellant $67.1%$
- Second stage $24.9%$
Second stage:
- construction $8%$ ($2.0%$ from initial mass)
- Propellant $58.9%$ ($14.7%$ from initial mass)
- Final payload $33.1%$ $(8.3%$ initial mass)
So in TOTAL the shares are:
- $8.25%$ the final payload
- $81.76%$ propellant
- $9.99%$ construction
As the payload needs to be $40 \text t$, I end up with:
First stage:
- Construction $38.8 \text t$
- Propellant $325.2 \text t$
- Second stage $120.8 \text t$
TOTAL: $484.8 \text t$
- Second stage:
- Construction $9.7 \text t$
- Propellant $71.1 \text t$
- Final payload $40 \text t$
TOTAL $120.8 \text t$
So far so good (even though I am skipping many details as this is not really my field of science at all, I am trying to keep the focus of only getting the order of magnitude of the energy use...)
Now I am in trouble though... Which energy equation should I use:
$$E = 0.5 \times (m_0-m_1) \times Ve^2$$ ($m_0-m_1$ would be the mass of the propellant)
or
$$E = 0.5 \times m_1 \times (e^{\Delta V/V_e}-1) \times V_e^2$$
With the first equation I get $3293 \text {GJ}$ and $0.720 \text {GJ}$, so $4013 \text {GJ}$ in TOTAL.
With the second I got confused with m1. In first stage the $m_1$ is the mass of the second stage ($120.8 \text t$)? But with the second stage it must be the mass of the payload and the construction, right...? Or just the payload ($40 \text t$)? Or is this matter of definition, how loose boundaries do I want to set to these calculations? (the payload is ONLY the useful payload ($= 40 \text t$) or also the whole construction of the second stage (= useful payload + engines, empty tanks etc)
Thanks a lot, if there's still more people willing to help in this interdisciplinary research of mine...