Answer to Question #89382 in Astronomy | Astrophysics for sathwik

Question

For a special mission to Mars you need to know the smallest distance between Earth and Mars.
However, you have lost your astronomy book and you could only find these values:
Distance Earth to Sun: 149.6 million km
Orbital period Earth: 1.00 years
Orbital period Mars: 1.88 years
By using these values and assuming that Mars and Earth move an circular orbits, calculate the
smallest possible distance between Earth and Mars.
Expert's answer

Answer 1

Assuming that both Mars and Earth move along the circular orbits (within the same plane), the minimal distance occurs when both planets are aligned within the common radial line. Hence, its value can be obtained as

$$\Delta R = R_M - R_E$$

The RM value can be obtained, for instance, by applying the Newton's universal law of gravitation to both planets (below M is the mass of the Sun):

$$G \frac{M m_E}{R_E^2} = m_E a_E, \quad a_E = \frac{4 \pi^2 R_E}{T_E^2} \, \Rightarrow \, GM = \frac{4 \pi^2 R_E^3}{T_E^2}$$

The same expressions are valid for Mars (one has to change index "E" to "M" only). Hence, we obtain:

$$\frac{R_E^3}{T_E^2} = \frac{R_M^3}{T_M^2} \, \Rightarrow \, R_M = R_E \left(\frac{T_M}{T_E}\right)^{2/3}$$

Finally,

$$\Delta R = R_E \left[ \left(\frac{T_M}{T_E}\right)^{2/3} - 1\right]$$

Substituting the numerical values, we obtain:

$$\Delta R = 149.6 \cdot \left[ \left(\frac{1.88}{1.00}\right)^{2/3} - 1 \right] \approx 78.3 \, million \, km$$

Answer: 78.3 million km.