Mathematically Modeling NASA’s Upcoming Insight Mission to Mars
Introduction:
Ever since I was a little boy I’ve been fascinated with space. Being an astronaut would be out of this world; however, mechanical engineering has always been a dream of mine. Rockets and satellites are of great interest to me, and the future Mars missions are exciting. As I can't physically travel to Mars, the pictures and data collected by the many NASA probes sent there are the next best thing. At first, I thought that sending a rocket to Mars wouldn’t be that hard; you just aim the rocket at Mars and launch it. However, it is much more complex than that. In my investigation I will mathematically model the next rocket launch to Mars which is NASA’s Insight
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The orbits of Earth and Mars are not perfect circles, and as a result their distance from the sun varies. The average distance from the sun will be used in this investigation. The average distance of Earth from the sun is 1AU or 149,600,000,000m, while the average distance of Mars from the sun is 1.52AU or 227,940,000,000m (Universe Today). These points can be used to find the semi-major axis a=(1+1.52)/2 a=1.26 AU
Using Kepler's 3rd law, the time it would take to reach Mars can be calculated. Using the equation p²=a³ where p is the period of the transfer orbit. p=√1.26³ p=1.41years
Given that the period for the elliptical transfer orbit is 1.41 years, the time to reach Mars would be .705 years or half of the transfer orbit. 0.705 years is equal to roughly 257 days. To calculate the velocities needed for the orbit, the following equations are used:
V₁=√GM/r₁
V₂=√GM/r₂
Vp=V₁+ΔV₁
V₂=Va+ΔV₂
GM=the gravitational constant x mass of the sun or (6.687x10-11) x (1.989x1030) which equals 1.327x1020
Vp=velocity of perigee
Va=velocity of apogee
To calculate the ΔV that is required the equations for the conservation of angular momentum and the conservation of energy
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However, by having a smaller change in velocity, the trip requires less fuel and costs less as a result. When comparing this hypothetical journey to Mars to NASA's planned mission, the orbital path is similar; however, the time to arrive is longer by about 52 days. Although not exactly the same flight duration, the difference can be a result of the fact that Earth and Mars’ orbits aren't perfect circles and have varying distances from each other which affect travel distance and times. The difference in time can also stem from Kepler's Third Law: it only factors in the semi-major orbit and the effect of the sun's gravity to determine speed. However, there was no way to factor in the speed of the rocket to see its affect on the time. In conclusion, the Hohmann Transfer Orbit is the method of choice for spacecraft and offers reliable, albeit not 100 percent accurate data of the time and changes in velocity for an interplanetary
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