I was watching a video that displayed how the time to get to Mars varied with the production of varying delta-v and was looking for the mathematics behind such a calculation. Here is the link to the video:
We can make some high-level assumptions to understand the approximate time taken to reach Mars orbit as a function of the delta-v applied at Earth.
First, we will assume that the orbits of Earth and Mars are both perfectly circular and co-planar. The assumed properties of each orbit is given in the table below:
We can then consider a simple model since we are interested in approximate numbers: A purely impulsive delta-v is applied at some point from the Earth orbit in the velocity direction which will put our spacecraft on a Mars crossing trajectory. We then want to obtain the time taken following the application of the impulsive delta-v to cross Mars' orbit around the sun.
Note that for this simple model we will forget about the phasing of the Earth departure and Mars arrival and will assume that the timing of the transfer will be selected to ensure Mars is at the crossing point at the same time as the spacecraft.
We then need to calculate the time to reach the crossing position as a function of delta-v applied. The steps to do this are roughly as follows:
- Add the delta-v applied to the circular velocity of Earth's orbit to get the velocity at periapsis for the transfer orbit.
- Get the Semi Major Axis of the transfer orbit from the specific orbital energy equation using the values from step 1.
- Using the Semi Major Axis obtain the radius of Apogee and the Eccentricity of the transfer orbit.
- Evaluate the Mean Eccentricity at the Mars orbit crossing location. To do this first obtain the True Anomaly at this position from the Semi Major Axis and Eccentricity. Then by using the True Anomaly you can derive the Eccentric Anomaly, from which you can obtain the Mean Anomaly.
- Calculate the mean motion of the orbit.
- We can obtain the flight time between two points in the orbit by finding the difference between the Mean Anomaly at the two points and dividing this value by the mean motion. Since we start from perigee we can get the time of flight by simply dividing the Mean Anomaly at the Mars crossing point by the mean motion.
Using these steps you can create a plot showing time of flight as a function of delta-v applied.
Again, this is only a simple analysis to get some rough numbers. The scenario described is not realistic since the orbits of Earth and Mars are not perfectly spherical and co-planar. However, hopefully this helps enough to give some approximate values and some insight into how these numbers can be analytically obtained.
To get more accurate picture you can look into obtaining real ephemeris data for Earth and Mars coupled with a Lambert Solver. If you know a little python I highly recommend this video series on YouTube to get an idea of how this can be done: