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.

Answer 1

Assuming circular orbits for both Mars and Earth, one can deduce that minimal distance can be observed when both planets are aligned within the common radial line:

$$\Delta R = R_M - R_E$$

Applying the 2nd Newton's law to the planetary motion, we obtain:

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

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

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

As a result,

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

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.