Question

The orbit of the Earth about the Sun is almost circular. The closest and farthest distances are $1.47\times {10}^8\mathrm{km}$ and $1.52\times$ ${10}^8\mathrm{km}$, respectively. Determine the maximum variations in ( $a$ ) potential energy, (b) kinetic energy, ( $c$ ) total energy, and (d) orbital speed that result from the changing Earth-Sun distance in the course of 1 year. (Hint: Use conservation of energy and angular momentum.)

Short answer

This question requires computing different forms of energies and comparing them at different Earth-Sun distances. The process demands the computational knowledge in physics, understanding of the conservation of energy, gravitational potential energy and conservation of angular momentum combined with kinetic energy equations.

Step-by-step solution

Calculate potential energy

To find the potential energy, use the formula for gravitational potential energy: $$\text{Potential Energy (PE)}=-\frac{GMm}{r}$$whereG = gravitational constant = $6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$,M = solar mass = $1.99\times {10}^{30}\text{kg}$,m = Earth's mass = $5.98\times {10}^{24}\text{kg}$,r = distance between Earth and the Sun.Evaluate this expression for both minimum and maximum distances to calculate the difference or variation. Be sure to convert kilometers to meters, as G is given in m^3/kg/s^2.

Determine kinetic energy

To determine kinetic energy, first remember that total energy (E) is conserved and is given as $$E=\text{KE}+\text{PE}=-\frac{GMm}{2a}$$from which $$\text{KE}=E-\text{PE}=-\frac{GMm}{2a}-(-\frac{GMm}{r})=\frac{GMm}{2a}-\frac{GMm}{r}$$wherea = semi-major axis = $\frac{r_{\text{min}}+r_{\text{max}}}{2}.$Again calculate for both situations (closest and farthest Earth-Sun distances) and find the difference.

Calculate total energy

The total energy variation is just the sum of potential and kinetic energy variations.

Calculate orbital speed

For the orbital speed, use that the angular momentum is conserved, i.e., $$m{v}_{\text{min}}{r}_{\text{min}}=m{v}_{\text{max}}{r}_{\text{max}}$$to find $$v_{\text{max can be found from}}\frac{v_{\text{min}}{r}_{\text{min}}}{r_{\text{max}}}$$and then calculate $v_{\text{min}}$ using the kinetic energy equation $KE=\frac{1}{2}m{v}_{\text{min}}^2.$ Then calculate the difference in speeds.

Key concepts

Gravitational Potential Energy

In the context of Earth's orbit around the Sun, gravitational potential energy is a key concept. Potential energy (PE) between two masses, such as Earth and the Sun, is defined by the formula $$\text{Potential Energy (PE)}=-\frac{GMm}{r}$$where:

• $G$ is the gravitational constant $6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$
• $M$ is the mass of the Sun $1.99\times {10}^{30}\text{kg}$
• $m$ is the mass of the Earth $5.98\times {10}^{24}\text{kg}$
• $r$ is the distance between Earth and the Sun in meters

Potential energy is negative, indicating a bound system where Earth is gravitationally bound to the Sun.
The potential energy varies inversely with distance; that means when the Earth is closer, potential energy becomes less negative (increases), and vice versa. Evaluate the expression for the minimum and maximum distances to find the variation in potential energy during Earth's orbit.

Kinetic Energy

Kinetic energy (KE) is related to the motion of Earth in its orbit. It can be calculated by considering the conservation of energy in Earth's nearly circular orbit. The formula for kinetic energy is: $$\text{KE}=E-\text{PE}$$Here, $E$ is the total energy which is conserved. Total energy is expressed as $$E=\text{KE}+\text{PE}=-\frac{GMm}{2a}$$where $a$ is the semi-major axis, the average of the closest and farthest distances. To find the kinetic energy at different points, use:$$\text{KE}=\frac{GMm}{2a}-\frac{GMm}{r}$$This allows us to see how kinetic energy changes as Earth moves through different points in its orbit. When Earth is closer to the Sun, its velocity, and thus kinetic energy, is higher.
When it is farthest, the kinetic energy decreases. Calculate KE for both the minimum and maximum distances to determine the variation.

Conservation of Energy

The principle of conservation of energy plays a crucial role in understanding the Earth's orbit. According to this principle, the total energy of the system - which is the sum of kinetic energy and gravitational potential energy - remains constant over time in a closed system without any external forces.
In Earth's orbit, the primary assumption is that neither energy nor mass is added or removed. Therefore, the total mechanical energy (E) is conserved:$$E=\text{KE}+\text{PE}=\text{constant}$$This means that when potential energy increases due to Earth's position being further from the Sun, kinetic energy decreases to maintain a constant total energy, and vice versa. Understanding this balance can help explain how Earth's speed varies with distance from the Sun, giving rise to the concept of equilibrium in dynamic systems.

Orbital Speed

Orbital speed is the velocity at which Earth travels around the Sun. It is influenced by the gravitational forces acting upon it and varies as Earth moves from its closest point (perihelion) to the farthest point (aphelion) in its orbit. The relation between speed and position in the orbit can be understood using the conservation of angular momentum:Given by the equation:$$m{v}_{\text{min}}{r}_{\text{min}}=m{v}_{\text{max}}{r}_{\text{max}}$$This tells us that the product of the mass, velocity, and distance from the center of rotation remains constant. By rearranging, the maximum speed $v_{\text{max}}$ is found from:$$v_{\text{max}}=\frac{v_{\text{min}}{r}_{\text{min}}}{r_{\text{max}}}$$To find $v_{\text{min}}$, use:$$KE=\frac{1}{2}m{v}_{\text{min}}^2$$Kinetic energy helps in deducing the velocity at different points of the orbit. As Earth moves closer, it speeds up, and as it moves away, it slows down, illustrating the captivating dance of celestial mechanics.