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For the second part of your question, local acceleration does matter for the thrust-to-weight ratio. Isp is exhaust velocity where $g_0$ is divided out by arbitrary convention, your rocket engine performing the same in any gravitational environment. I agree with Christopher James Huff that just formulating the rocket equation in terms of exhaust velocity ...


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g0 is indeed an idealized "surface gravity" for Earth. As arbitrary and nonsensical as that is for many applications, you may be better off just directly using the effective exhaust velocity: $$\Delta v=v_{e} \ln \left(\frac{m_{0}}{m_{f}}\right)$$


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