Place the following in order from largest to smallest semimajor axis. a. a planet with a period of 84 Earth days b. a planet with a period of 1 Earth year c. a planet with a period of 2 Earth years d. a planet with a period of 0.5 Earth year

The semimajor axis is a crucial part of an orbit's geometry. It is the longest radius of an elliptical orbit, extending from the center of the ellipse to its furthest edge. Essentially, it's half of the longest diameter across the orbit.
For planets orbiting the sun, the semimajor axis determines the size of the orbit and is directly related to the orbital period. The longer the semimajor axis, the longer it takes for a planet to complete one full orbit. In simpler terms, a larger semimajor axis means a larger pathway around the sun and thus, a longer journey. For Kepler's Third Law, this axis is represented by 'a'.
Understanding the semimajor axis helps explain why planets with longer orbital periods are further from the sun.

The orbital period is the time a planet takes to complete one full orbit around the sun. It is a fundamental aspect of planetary motion studied in Kepler's laws. In our exercise, the orbital periods of planets were given in Earth days and Earth years.
To use Kepler's Third Law effectively, we first converted each period to Earth years. This allowed us to uniformly apply the law and compare the semimajor axes.
Kepler's Third Law states the square of the orbital period (T) is proportional to the cube of the semimajor axis (a):

This means planets with longer periods (like 2 Earth years) will have a larger semimajor axis compared to those with shorter periods (like 84 Earth days or 0.23 Earth years).

Planetary orbits often seem circular, but they are actually ellipses. Kepler's First Law tells us this. Each planet moves in an elliptical orbit with the sun at one of two focal points.
Understanding the shape and size of these orbits is key. While the shape is defined by its eccentricity, the size comes from the semimajor axis, which we discussed earlier. Planets in larger orbits will travel at slower speeds in their paths due to the larger distance from the sun.
Moreover, different planets orbit at different distances and speeds due to the variation in their semimajor axes. Longer semimajor axes result in more extensive orbits and longer orbital periods. This is why outer planets like Neptune take much longer to orbit the sun compared to inner planets like Mercury.

An important aspect of comparing astronomical quantities is understanding the order of magnitude. This term refers to the scale or size of a number. When comparing semimajor axes or orbital periods, the order of magnitude helps us quickly estimate and compare values.
For instance, in our problem, converting periods to Earth years ranged from 0.23 to 2 Earth years. We used the order of magnitude to apply Kepler's Third Law and subsequently found each resulting semimajor axis.
Determinino the order of magnitude helps simplify complex astronomical calculations and highlights the relative differences in size and scale between planets' orbits.