Question

Mars orbits the Sun at a mean distance of 228 million \(\mathrm{km},\) in a period of 687 days. The Earth orbits at a mean distance of 149.6 million \(\mathrm{km},\) in a period of 365.26 days. a) Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line? b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth. c) Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

Short answer

What is the time between two closest approaches of Earth and Mars? What is the angle between lines drawn in two closest approach situations? Solution: a) To calculate the angle between Earth-Sun and Mars-Sun lines after one Earth orbit, first find the fraction of Mars' orbit by dividing the time it takes Earth to complete one orbit (365.26 days) by the time Mars takes to complete one orbit (687 days). Next, find the angles for both Earth and Mars and calculate their difference. b) To find the time between two closest approaches, determine how many times the angle difference must be covered to complete a full circle (360 degrees). Multiply this number by 365.26 days to find the synodic period. c) To find the angle between lines in two closest approach situations, calculate the number of times the angle difference is formed during the synodic period and multiply that number by the angle difference.

Step-by-step solution

a) Finding the angle between Earth-Sun line and Mars-Sun line after one Earth year

First, we need to find out how much each planet has orbited the Sun after 365.26 days. For Mars, we can find the fractional Mars orbit covered relative to the Earth's 365.26 days. Fraction of Mars orbit = Time_Earth / Time_Mars = 365.26 days / 687 days. Now we need to find the angle corresponding to that fractional orbit for both Mars and Earth. Angle_Earth = 360 degrees (As Earth completes one orbit) Angle_Mars = Fraction of Mars orbit * 360 degrees. The angle between Earth-Sun line and Mars-Sun line after one Earth year is the difference in their respective angles: Angle_difference = Angle_Earth - Angle_Mars.

b) Finding the time between two closest approaches

Now that we have the angle formed after one Earth year, we can find out how many times this angle must be covered to reach 360 degrees, which represents a complete circle. Number_of_times = 360 degrees / Angle_difference To find the time between two closest approaches (synodic period), we multiply the number_of_times by 365.26 days. Synodic period = Number_of_times * 365.26 days.

c) Finding the angle between lines drawn in two closest approach situations

The angle_between_lines is already calculated in part (a). Thus, we can use the same procedure to find the number of times this angle is formed in the synodic period. Number_of_times_synodic = Synodic period / 365.26 days Angle in two closest approaches = Number_of_times_synodic * Angle_difference.

Key concepts

Synodic Period

The synodic period is a critical concept in the study of orbital mechanics, particularly when examining interactions between two celestial bodies orbiting a common focus, such as the Sun. It refers to the time it takes for two bodies to return to the same or a similar relative configuration or alignment — essentially, it's the time between two consecutive similar conjunctions. For instance, the synodic period of Earth and Mars is the time between two closest approaches of the two planets.

The calculation of the synodic period is rooted in understanding the differing orbital periods of the bodies involved. Earth takes roughly 365.26 days to complete its orbit around the Sun, whereas Mars completes its orbit every 687 days. These two different periods mean both planets are continually shifting relative to each other. As students work with these concepts, it becomes evident that the synodic period is not simply the average of the two orbital periods; rather, it's the timeframe in which Earth (or another planet) 'catches up' and realigns with the other body after both have completed whole fractions of their orbits.

Real-World Application

• Planning interplanetary missions, where missions are often launched during periods of closest approach to minimize travel time and fuel requirements.
• Understanding and predicting celestial events, such as eclipses, transits, and conjunctions, which depend on the relative positions of celestial bodies.

Planetary Orbits

The orbits of planets around the Sun are not just paths; they are the key to understanding the movement of celestial bodies in our solar system. When visualizing these orbits, it can be helpful to think of them as circular tracks, each with a different radius and each planet moving at a particular rate. Earth orbits at a distance of about 149.6 million kilometers, while Mars's mean distance is roughly 228 million kilometers from the Sun.

One might simplify the motion by imagining Earth and Mars as runners on separate tracks, each completing laps in their own time — Earth's lap taking approximately 365.26 days and Mars's lap 687 days. This analogy helps highlight that they are racing at different speeds due to the distances from the central sun, which is key to solving many orbital challenges.

Key Points for Consideration

• The orbits' circular simplification aids in problem solving, despite the actual elliptical nature of orbits.
• Understanding the differences in orbital radius and period is essential for calculating positions of the planets relative to each other over time.

Angular Displacement

Angular displacement in orbital mechanics is a measure of the angle through which a planet rotates about the Sun in a given period. It's analogous to the distance covered on a circular track but measured in degrees (or radians) rather than meters or kilometers.

In our problem with Earth and Mars, we're particularly interested in the angular displacement of each planet after a set period — such as a year on Earth. This concept helps us determine how far along it's orbit a planet is, or in other words, its position relative to the Sun and other planets at a given time. That angular displacement is what sets the stage for calculating when two planets like Earth and Mars will be in the closest proximity, relevant for both observational astronomy and space mission planning.

Real-World Implications

• It is used to predict planetary positions, which is essential for various astronomical applications, including creating calendars and navigation charts.
• In spacecraft navigation, calculating angular displacement aids in precise trajectory plotting for interplanetary travel.