Show that Eq. (1.20) reduces to the usual simple form of...

Show that Eq. $(\underline{1.20})$ reduces to the usual simple form of Kepler's third law for planetary motion $\left(P^2=a^3\right)$ if time is measured in years, masses in units of $M_{\odot}$, and distances in $\mathrm{A}_{\mathrm{U}}$, and the mass of a planet is neglected relative to that of the Sun.***

Key Concepts

Kepler's Third Law

Kepler's Third Law establishes the relationship between the orbital period of a planet and the size of its orbit, stating that the square of the period (P²) is proportional to the cube of the semi-major axis (a³). This principle provides a fundamental law for planetary motion, and under the assumption that one mass greatly exceeds the other (i.e., the Sun versus a planet), the relation simplifies to a universal form with a constant of proportionality of unity in appropriate units.

Dimensional Analysis and Unit Conversion

Dimensional analysis ensures that the physical quantities in an equation are expressed in a consistent unit system. When time is measured in years, distance in astronomical units, and mass in solar masses, the equation naturally simplifies due to the chosen system of units. This conversion absorbs factors such as the gravitational constant into the unit definitions, leading to the simplified Kepler's law form without extra constants.

Neglecting the Planet’s Mass

In the context of planetary motion, the mass of a planet is significantly smaller than that of the Sun. By neglecting the planet's mass, the two-body problem reduces to a one-body problem where the gravitational field is essentially determined by the Sun. This simplification is crucial in deriving the simple form of Kepler's Third Law, as it allows the constant of proportionality to depend solely on the Sun's mass.

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