Question

Express the universal gravitational constant \(G\) that appears in Newton's law of gravity in terms of the astronomical unit \(\mathrm{AU}\) as a length unit, the solar mass \(M_{\mathrm{S}}\) as a mass unit, and the year as a time unit. \(\left(1 \mathrm{AU}=1.496 \times 10^{11} \mathrm{~m}, 1 \mathrm{M}_{\mathrm{S}}=1.99 \times\right.\) \(\left.10^{30} \mathrm{~kg}, 1 \mathrm{y}=3.156 \times 10^{7} \mathrm{~s} .\right)(b)\) What form does Kepler's third law (Eq. \(14-23\) ) take in these units?

Short answer

The Universal Gravitational Constant in these units is \(G = 4 \pi^2 \mathrm{AU}^3 / (\mathrm{M}_\mathrm{S} \mathrm{y}^2)\). Kepler's Third law takes the form \(T^2 = a^3\) in the given units.

Step-by-step solution

Definition of Universal Gravitational Constant (G)

We are aware that the Universal Gravitational Constant, \(G\), is given by the expression \(G = F r^2/(m_1 m_2)\), where \(F\) is the gravitational force between two objects, \(r\) is the distance between the objects, and \(m_1\) and \(m_2\) are the masses of the objects.

Convert the units of the gravitational constant

Given that we have \(M_{\mathrm{S}}=1.99 \times 10^{30} \mathrm{~kg}\) which represents the mass of the Sun, \(r = 1 \mathrm{AU}=1.496 \times 10^{11} \mathrm{~m}\) which is the distance between the Earth and the Sun, and \(F = M_{\mathrm{s}} a = M_{\mathrm{S}} \cdot 4 \pi^2/(1 \mathrm{y})^2 = 1.99 \times 10^{30} \mathrm{~kg} \cdot (4 \pi^2/(3.156 \times 10^{7} \mathrm{~s})^2)\) which is the gravitational force between Earth and the Sun, we can substitute these values into the equation for \(G\) to get \(G = 4 \pi^2 \mathrm{AU}^3 / (\mathrm{M}_\mathrm{S} \mathrm{y}^2)\).

Formulate Kepler's third law in the given units

Kepler's third law states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit. This can usually be expressed using the relation \(T^2 = K a^3\) where \(K\) is a constant. In the natural units presented, the gravitational force that determines orbital motion involves only one solar mass and the distance is measured in astronomical units, \(K = 1\). Thus, the Kepler's Third law takes the form \(T^2 = a^3\) in the given units.

Key concepts

Universal Gravitational Constant

The universal gravitational constant, denoted as \( G \), is a key quantity in Newton's law of gravity. This constant essentially governs the strength of the gravitational force between two masses. According to the law, the force \( F \) between two objects is directly proportional to the product of their masses \( m_1 \) and \( m_2 \), and inversely proportional to the square of the distance \( r \) between them, described by the formula \( F = G \frac{{m_1 m_2}}{{r^2}} \).

Understanding \( G \) in terms of other units is crucial for calculations in astronomy. For instance, if we use astronomical units (AU) for distance, the solar mass (\( M_{\mathrm{S}} \)) for mass, and years for time, we can express \( G \) in a form that's more practical for studying planetary motions within our solar system. By translating \( G \) into these commonly used astronomical units, students can better understand gravitational interactions on a scale that's beyond everyday experiences. This also allows for simplifying complex equations and leads to more intuitive comprehension of celestial mechanics.

Astronomical Unit

An astronomical unit (AU) is a unit of measurement that is standardly used to describe distances within our solar system. Precisely, 1 AU is defined as the average distance between the Earth and the Sun, approximately \(1.496 \times 10^{11}\) meters. This unit simplifies celestial calculations by relating vast interplanetary distances to a more manageable unit based on an Earth-centric perspective.

Utilizing the AU helps students conceptually transition from thinking about distances on earthly scales to grasping the expanse between celestial bodies. For example, the distance of a planet from the Sun expressed in AUs can instantly tell you how many times further it is compared to the Earth-Sun distance. This is especially helpful when applying concepts like Kepler's third law, which relates the distance of a planet from the Sun to its orbital period.

Kepler's Third Law

Kepler's third law of planetary motion states that the square of the orbital period \( T \) of a planet is directly proportional to the cube of the semi-major axis \( a \) of its orbit. This is generally expressed as \( T^2 = Ka^3 \), where \( K \) is a proportionality constant dependent on the system's specific characteristics. When applied using astronomical units for distance and years for time, the constant \( K \) becomes 1, simplifying the law to \( T^2 = a^3 \).

This elegant form of Kepler's third law, when described with AU and years, enables clear and direct comparisons between the orbital periods and distances of different planets from the Sun. It does so without the need to include the complexities that arise from varying units of measurement. Through this law, students can appreciate the ordered nature of our solar system, where the seemingly abstract motion of celestial bodies is governed by a beautifully simple mathematical relationship.