Due 12/16/16 Aero 201 Project 3 Professor Ken Powell, 3052 FXB (Aero), powelloum

Question

Due 12/16/16 Aero 201 Project 3 Professor Ken Powell, 3052 FXB (Aero), powelloumich.edu Problem Description In this project, you will use analytical and numerical tools to design a mission to Mars. t 0 state Mars Sun Earth Figure 1: Simplified Solar System To simplify the analysis, you will only consider a subset of the solar system, consisting of the Sun, Mars, and Earth. Place the Sun at the origin, and assume circular orbits, all in the same plane, for the Earth and Mars about the Sun. Relevant parameters are given in the table below Object Orbit radius (m) Mass (kg) Object radius (m) 23 3.389 x 10 Mars 2.2789 x 10 6.390 x 10 Earth 1.4958 x 10 5.972 x 1 24 6.371 x 1 11 1.989 x 1030 Sun Use G 6.6710 kgs for the gravitational constant, and assume that the Sun we much more than the Earth and Mars, so that Au is just G times the mass of the central body. Both Earth and Mars move in counterclockwise orbits around the Sun in the z, y plane. At time t 0, the Earth and Mars are 90° apart in their respective orbits, which means that

Explanation / Answer

To calculate the semi major axis a= (r1+r2)/2

where r1= orbital radius of earth(perihelion)=1 AU

r2=orbital radius of mars(aphelion)=1.524AU

Now semi major axis a=(1+1.524)/2

=2.524/2

=1.262 AU

Eccentricity of the transfer orbit e=1-r( perihelion)/a

r perihelion or radius of earth=1AU

a=semi major axis

e=1-1/1.262

=1-0.792

=0.208

Now orbital period required to travel from earth to Mars

According to kepler's third law

p2=a3

a= semi major axis=1.262

p= orbital period=??

p=(1.262)3/2

p=1.417 years=517 days

Since it take around half of the time to travel to Mars

therefore 517/2=258.5=259 days

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