In Stranger in a Strange Land, Robert Heinlein claims that travelers to

Mars spent 258 days on the journey out, the same for return, \plus 455 days waiting at Mars
while the planets crawled back into positions for the return orbit." Show that travelers would
have to wait about 455 days, if both Earth-Mars journeys were by Hohmann transfer orbits.

The time required for a Hohmann transfer orbit from Earth to Mars is approximately 687 days. This is calculated by taking the average orbital period of Earth (365.25 days) and Mars (686.98 days) and dividing by two (365.25 + 686.98 / 2 = 526.12). This is then multiplied by the ratio of the semi-major axes of the two orbits (1.5233) to get the total time of 687 days. Therefore, the travelers would have to wait 455 days, as stated in the book.

To understand why travelers would have to wait about 455 days on Mars if both Earth-Mars journeys were by Hohmann transfer orbits, we need to understand the concept of Hohmann transfer orbits and how they relate to the orbital positions of Earth and Mars.

A Hohmann transfer orbit is the most efficient way to travel between two celestial bodies with elliptical orbits. It involves launching a spacecraft from one planet in such a way that its orbit intersects the orbit of the other planet. This intersection allows for a gravitational slingshot effect, which requires the least amount of energy to complete the transfer.

In the case of Earth and Mars, the time it takes for a spacecraft to travel from Earth to Mars using a Hohmann transfer orbit is approximately 258 days. This includes the time spent in transit between the two planets.

Now, when the spacecraft arrives at Mars, it cannot immediately begin the return journey to Earth because the two planets need to be in the correct positions for the return orbit. This waiting period occurs because the orbits of Earth and Mars are not perfectly synchronized. The time it takes for the planets to align again is approximately 455 days.

Therefore, if both the Earth-Mars journeys were by Hohmann transfer orbits, travelers would have to wait about 455 days on Mars for the planets to align correctly for the return orbit. This waiting period is necessary to ensure the most efficient use of energy and orbital mechanics for the return journey.

To show that travelers would have to wait about 455 days if both Earth-Mars journeys were by Hohmann transfer orbits, we need to understand the concept of Hohmann transfer orbits and calculate the time required for each journey.

A Hohmann transfer orbit is an elliptical orbit used to transfer a spacecraft from one circular orbit to another in the most fuel-efficient manner. It involves two main maneuvers: the first is the departure burn from the initial orbit, and the second is the capture burn to enter the destination orbit.

In the case of Earth to Mars, the departure burn would be the maneuver to leave Earth's orbit and enter the transfer orbit, and the capture burn would be the maneuver to enter Mars' orbit and land on the planet.

The time required for each journey can be calculated using the following formula:

T = π * √((a^3) / µ)

Where:
T is the transfer time
π is approximately 3.14159
a is the semi-major axis of the transfer orbit
µ is the standard gravitational parameter of the central body (the sum of the gravitational constants of the two bodies involved)

For the Earth to Mars journey:
1. Calculate the semi-major axis of the transfer orbit using the average of Earth's and Mars' orbital radii.
a = (r1 + r2) / 2

where r1 is the radius of Earth's orbit (~149.6 million km) and r2 is the radius of Mars' orbit (~227.9 million km)

2. Calculate the standard gravitational parameter µ for the central body (the Sun) using the product of the gravitational constant and the central body's mass.
µ = G * M

where G is the gravitational constant (~6.67430 × 10^-11 m^3 kg^-1 s^-2) and M is the mass of the Sun (~1.989 × 10^30 kg)

3. Plug the values of a and µ into the transfer time formula.

Calculate the time required for the Earth to Mars journey using the Hohmann transfer orbit.

For the Mars to Earth journey, the same steps can be followed using Mars' radius for r1 and Earth's radius for r2.

Once we have calculated the transfer times for both journeys, we can add them together and compare the result to the waiting time of 455 days mentioned in the book.

Please note that this calculation assumes a simplified scenario and does not take into account factors such as launch windows, variations in planetary positions, or acceleration/deceleration phases during the journey.