Difference in using flat earth or round earth model for equation of motion

Question

according to the book of Mr. Peter H. Zipfel (Modeling and simulation of aerospace vehicles) for high speed or/and long flight applictions kinematic simulations should be done by round rotating earth models. My question: why is it not correct to use flat earth models? Where is the difference?

Rotation vs no rotation: I assume that the "rotation" only plays a role for calculating radial forces (centrifugal, coriolis). Because for the position of the vehicle it does not change anything if the earth rotates or not, because due to the rotation of the atmosphere, vehicle is moving with the rotation of the earth.

Round vs Flat: The flat earth vs. round earth model should also only play a role in using the radial forces as well? Or any other effect? In a flat model the vehicle flies in a straight line, in a round earth model in a circular trajectory where radial forces act.

What do you think?

Thanks for your thoughts!

Answer 1

Spherical geometry is quite different than cartesian trigonometry.. the sum of all angles in a triangle can be up to 540°, distances are different etc. particular the initial choice of flight angle w.r.t north over long distances is not what you get from a flat model.

Answer 2

I'll dip my toe into the quicksand and try not fall in.

Spaceflight math dates back to the era of pocket protectors and slide rules. If you needed more digits to say multiply or take a square root, then you'd grab a table of logarithms and add the two logs or divide the log by two, and then find the antilog.

To calculate an approximate answer was often good enough (you were envelope-backing or there would be correction maneuvers later) so you dropped the small effects.

If your orbital period is 90 minutes, or only 4° per minute of rotation around the Earth.

Those more knowledgable than I will be able to answer with which effects would be most sensitive to the difference between say a minute of straight flight parallel to a flat Earth or curved flight parallel to a round Earth.

Even for an 8 minute launch from Earth to orbit, you can calculate a flat Earth control program (thrust and tilt vs time) and add make only a minor increase in the rate of title to get it to insert into a circular orbit around a round Earth.

See for example the discussion of a linear tangent steering law in @RussellBorogove's answer to For an Apollo Lunar Module Ascent Stage launch, what is the optimal profile of 𝛽 (or 𝛾) vs time? which begins

Given some simplifying assumptions (constant thrust, constant gravitational field, flying in vacuum, over flat horizontal surface), which aren't too badly undermined by the lunar ascent case, the linear tangent steering law is known to give optimal orbital insertion:

$$\tan \theta = A \cdot t + B$$

and

• Coefficients for linear tangent steering law

These days one just writes a program and calls functions that evaluates the real shapes of Earth's surface and gravity field and lets it iterate to find the best flight path. But in the past that didn't happen in the classroom, or on the backs of envelopes or on note pads on engineers' desks.