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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:

Can Nuclear propulsion take us to Mars?

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    $\begingroup$ I think you're looking for Lambert's problem. Given two position vectors and a time of flight, a Lamberts solver will give you the velocity vectors at the two position vectors $\endgroup$ Dec 20, 2021 at 2:08
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    $\begingroup$ @AlfonsoGonzalez Is there an online calculator as such to make these calculations? $\endgroup$
    – Ryan
    Dec 20, 2021 at 7:20
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    $\begingroup$ I stopped watching the video at about the four minute mark based on excess spewed nonsense. Hohmann transfers are not used to get spacecraft to Mars. What is used is a solution to Lambert's problem, but then one needs a solution that makes sense. There are multiple questions and answers on this site that ask about pork chop plots (or porkchop plots), and also on the physics and astronomy sister sites. Unfortunately, developing those plots is computationally expensive, so you are not going to find an online tool that does that. Fortunately, python is more than capable of doing just that. $\endgroup$ Dec 20, 2021 at 12:39
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    $\begingroup$ @Ryan here is a simple online tool you can use to generate porkchop plots - though note that this is only approximate: sdg.aero.upm.es/index.php/online-apps/porkchop-plot $\endgroup$
    – Armadillo
    Dec 20, 2021 at 12:44
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    $\begingroup$ @Armadillo That's rather cool, and in Javascript no less. However, at some point, anyone interested in space exploration needs to know how to program. $\endgroup$ Dec 20, 2021 at 13:00

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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:

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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.

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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:

  1. Add the delta-v applied to the circular velocity of Earth's orbit to get the velocity at periapsis for the transfer orbit.
  2. Get the Semi Major Axis of the transfer orbit from the specific orbital energy equation using the values from step 1.
  3. Using the Semi Major Axis obtain the radius of Apogee and the Eccentricity of the transfer orbit.
  4. 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.
  5. Calculate the mean motion of the orbit.
  6. 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.

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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:

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    $\begingroup$ Great, exactly what I was looking for $\endgroup$
    – Ryan
    Dec 22, 2021 at 15:09
  • $\begingroup$ Also the amount of extra delta-v required does not scale linearly with the amount of extra velocity relative to Earth that you wind up with in solar orbit, assuming you're doing your departure burns from Earth efficiently. $\endgroup$
    – notovny
    Dec 22, 2021 at 15:33
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    $\begingroup$ Thank you for sharing my video! $\endgroup$ Dec 22, 2021 at 18:57

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