Question

a) By what percentage does the gravitational potential energy of the Earth change between perihelion and aphelion? (Assume the Earth's potential energy would be zero if it is moved to a very large distance away from the Sun.) b) By what percentage does the kinetic energy of the Earth change between perihelion and aphelion?

Short answer

After calculating GPE at both perihelion and aphelion, we get: GPE at perihelion: \(U_{peri} = -\dfrac{(6.674\times10^{-11})(1.989\times10^{30})(5.972\times10^{24})}{(0.983 \times 1.496\times10^{11})} = -3.664\times10^{33} J\) GPE at aphelion: \(U_{aph} = -\dfrac{(6.674\times10^{-11})(1.989\times10^{30})(5.972\times10^{24})}{(1.017 \times 1.496\times10^{11})} = -3.548\times10^{33} J\) #tag_title#Step 2: Calculate the KE at perihelion and aphelion#tag_content# To calculate the KE at perihelion and aphelion, we need to find the Earth's velocities at those points. Earth's velocity can be obtained using the following formula derived from the conservation of angular momentum: \(v = \sqrt{\dfrac{GM}{r}}\) Velocity at perihelion: \(v_{peri} = \sqrt{\dfrac{(6.674\times10^{-11})(1.989\times10^{30})}{(0.983 \times 1.496\times10^{11})}} = 2.978\times10^4 m/s\) Velocity at aphelion: \(v_{aph} = \sqrt{\dfrac{(6.674\times10^{-11})(1.989\times10^{30})}{(1.017 \times 1.496\times10^{11})}} = 2.926\times10^4 m/s\) Now, using the KE formula: \(K = \dfrac{1}{2}mv^2\) KE at perihelion: \(K_{peri} = \dfrac{1}{2}(5.972\times10^{24})(2.978\times10^4)^2 = 2.643\times10^{33} J\) KE at aphelion: \(K_{aph} = \dfrac{1}{2}(5.972\times10^{24})(2.926\times10^4)^2 = 2.573\times10^{33} J\) #tag_title#Step 3: Calculate percentage change in GPE and KE#tag_content# To find the percentage change in GPE and KE, we use the following formula: \(\dfrac{(\text{final value} - \text{initial value})}{\text{initial value}} \times 100\%\) Percentage change in GPE: \(\dfrac{(-3.548\times10^{33} - (-3.664\times10^{33}))}{-3.664\times10^{33}}\times 100\% = 3.17\%\) Percentage change in KE: \(\dfrac{(2.573\times10^{33} - 2.643\times10^{33})}{2.643\times10^{33}}\times 100\% = -2.64\%\) Short answer: Between perihelion and aphelion, the Earth's gravitational potential energy changes by 3.17%, while the kinetic energy changes by -2.64%.

Step-by-step solution

Calculate the GPE at perihelion and aphelion

To calculate the GPE at perihelion and aphelion, we first need to know the distances between the Earth and the Sun at these points. The average distance between Earth and Sun is 1 astronomical unit (AU), which is approximately \(1.496\times10^8\) km. At perihelion, the Earth is about 0.983 AU from the Sun, and at aphelion, the Earth is about 1.017 AU from the Sun. Now we can calculate the GPE at both these points using the formula: \(U = - \dfrac{GMm}{r}\), where \(G = 6.674\times10^{-11}m^3kg^{-1}s^{-2}\), \(M =1.989\times10^{30}kg\) and \(m=5.972\times10^{24}kg\).

Key concepts

Perihelion and Aphelion

The terms 'perihelion' and 'aphelion' refer to the points in the Earth's orbit around the Sun where the Earth is closest and farthest away, respectively. Understanding these concepts is crucial when considering variations in gravitational potential energy (GPE) and kinetic energy as the Earth moves around the Sun.

The Earth's orbit is not a perfect circle but an ellipse, so the distance between the Earth and Sun changes throughout the year. At perihelion, the gravitational pull of the Sun is strongest due to the shorter distance, causing the Earth to move faster. By contrast, at aphelion, the gravitational force is weaker, and the Earth's velocity decreases.

Importance of Perihelion and Aphelion

These points hold significance not only in calculating energy changes but also in understanding Earth's seasonal variations. For example, perihelion occurs in early January, around the same time as the northern hemisphere's winter, which may seem counterintuitive since being closer to the Sun would suggest warmer temperatures. However, seasons are primarily determined by the tilt of the Earth's axis rather than its distance from the Sun.

Astronomical Unit

When discussing distances in space, particularly within our solar system, we often use the term 'astronomical unit' (AU) for convenience. One AU is defined as the average distance between the Earth and the Sun, approximately 149.6 million kilometers (about 93 million miles). It provides a standard measurement that makes it easier to convey and comprehend the vast distances involved in astronomy.

Knowing that perihelion is about 0.983 AU and aphelion is about 1.017 AU helps us quantify the variation in distance in a more relatable way. Moreover, using the AU simplifies calculations when comparing distances across different points in Earth's orbit or when measuring distances to other astronomical bodies within our solar system.

Standardizing Space Measurements

By using the AU as a reference, we achieve a clearer understanding of how far objects are relative to each other, such as planets' positions or the extent of cometary orbits. It acts as a bridge between the unfathomable scale of space and more familiar units of measurement.

Kinetic Energy of Earth

The kinetic energy (KE) of the Earth is the energy it possesses due to its motion as it revolves around the Sun. This energy can be expressed using the equation KE = (1/2)mv^2, where 'm' is the mass of the Earth and 'v' is its orbital velocity. As the Earth travels from perihelion to aphelion, its velocity changes due to the variations in the gravitational force exerted by the Sun.

At perihelion, the Earth's kinetic energy reaches its peak because it's traveling at its fastest rate along the orbital path. Conversely, the kinetic energy is at its minimum at aphelion since the Earth slows down due to the greater distance from the Sun.

Energy Conservation in Orbital Motion

It's important to note that while the kinetic energy changes, the total energy of Earth's orbit, which is the sum of its kinetic and potential energy, remains constant if we ignore any external forces or energy loss mechanisms. This is an illustration of the conservation of mechanical energy in celestial mechanics. As the Earth moves in its elliptical path, its kinetic and potential energies continuously change in such a way that when one form of energy increases, the other decreases accordingly.