Problem 52 - University Physics with Modern Physics

Question

Sedna. In November \(2003,\) the now-most-distant-known object in the solar system was discovered by observation with a telescope on Mt. Palomar. This object, known as Sedna, is approximately 1700 \(\mathrm{km}\) in diameter, takes about \(10,500\) years to orbit our sun, and reaches a maximum speed of 4.64 \(\mathrm{km} / \mathrm{s}\) . Calculations of its complete path, based on several measurements of its position, indicate that its orbit is highly elliptical, varying from 76 \(\mathrm{AU}\) to 942 \(\mathrm{AU}\) in its distance from the sun, where \(\mathrm{AU}\) is the astronomical unit, which is the average distance of the earth from the sun \(\left(1.50 \times 10^{8} \mathrm{km}\right)\) . (a) What is Sedna's minimum speed? (b) At what points in its orbit do its maximum and minimum speeds occur? (c) What is the ratio of Sedna's maximum kinetic energy to its minimum kinetic energy?

Step-by-Step Solution

Answer

(a) 0.375 km/s. (b) Max at 76 AU, Min at 942 AU. (c) Ratio is 153.63.

Step 1: Understand Kepler's Laws

According to Kepler's laws of planetary motion, the speed of a celestial object varies depending on its distance from the sun due to the ellipticity of the orbit. This means the object's speed will reach its maximum when it is closest to the sun (perihelion) and its minimum when it is farthest (aphelion).

Step 2: Establish Sedna's Orbit Parameters

Sedna's closest distance (perihelion) to the sun is 76 AU, and its farthest distance (aphelion) is 942 AU. These represent the points where Sedna's speed will be at maximum and minimum, respectively.

Step 3: Apply Conservation of Angular Momentum

The conservation of angular momentum states: \[L = mvr = \text{constant}\]For Sedna, at perihelion and aphelion:\[v_{min} r_{max} = v_{max} r_{min}\]Thus:\[v_{min} = \frac{v_{max} r_{min}}{r_{max}} = \frac{4.64 \, \text{km/s} \times 76 \, \text{AU}}{942 \, \text{AU}}\]

Step 4: Calculate Minimum Speed

Convert the distances from AU to km (1 AU = \(1.50 \times 10^8\) km) and calculate:\[v_{min} = \frac{4.64 \times (76 \times 1.50 \times 10^8)}{942 \times 1.50 \times 10^8} = \frac{4.64 \times 76}{942}\]This simplifies to \[v_{min} \approx 0.375 \text{ km/s}\].

Step 5: Calculate Ratio of Kinetic Energies

Kinetic energy is given by \[KE = \frac{1}{2} mv^2\]. Hence, the ratio of maximum to minimum kinetic energy is:\[\frac{KE_{max}}{KE_{min}} = \left(\frac{v_{max}}{v_{min}}\right)^2\].Substitute the known values:\[\frac{KE_{max}}{KE_{min}} = \left(\frac{4.64}{0.375}\right)^2 \approx 153.63\].

Key Concepts

Orbital Mechanics

Orbital mechanics is a field of study that focuses on the motions of objects in space, specifically how celestial bodies orbit around each other. Sedna, a distant object in our solar system, exhibits these orbital characteristics as it dexterously weaves through space. Its movement provides a prime example of how the laws governing celestial motion work. The path of an object in space is often elliptical, as described by Kepler's Laws, indicating that the speed of the object will vary depending on its position along this path.
For Sedna, its orbit around the sun stretches from closest at 76 AU to farthest at 942 AU, showcasing how broad and elongated its elliptical trajectory is. Understanding such mechanics involves comprehending both gravitational forces acting upon the object and the energy relationships, which are crucial in determining speeds at different points along this orbit.

Conservation of Angular Momentum

The conservation of angular momentum is a vital principle in physics, indicating that if no external torque acts on a system, its angular momentum remains constant. For Sedna orbiting the sun, this principle suggests that the product of its velocity and distance from the sun remains unchanged as it travels through space. This means when Sedna is closer to the sun, it moves faster, and when it's farther, it moves slower.
Mathematically, angular momentum is expressed as \[L = mvr\] where \(m\) is the mass, \(v\) is the velocity, and \(r\) is the radius (distance from the sun). For Sedna at perihelion and aphelion:\[v_{min} r_{max} = v_{max} r_{min}\]. The constancy of \(L\) leads to Sedna's speeds being calculated at different orbit points, ensuring it follows its path with mesmerizing precision.

Astronomical Unit

The astronomical unit (AU) serves as a standard measure of distance within our solar system, equivalent to the average distance from the Earth to the sun, approximately \(1.50 \times 10^8\) kilometers. Understanding Sedna's orbit requires converting the orbital distances from AU to kilometers, revealing the vast expanses Sedna travels. For Sedna, its distances of 76 AU at its closest approach and 942 AU at its furthest highlight the immense scale of its orbit.
Using the AU as a measuring stick allows astronomers to more easily compare celestial distances and understand the sheer size of the pathways objects like Sedna take as they float through the cosmos.

Kinetic Energy Ratio

The kinetic energy of an object in motion, such as Sedna in its orbit, is described by the formula:\[KE = \frac{1}{2}mv^2\]. The ratio of maximum to minimum kinetic energy in Sedna’s orbit can paint a vivid picture of how much its speed and energy alter during its elliptic journey. This ratio is determined by:\[\frac{KE_{max}}{KE_{min}} = \left(\frac{v_{max}}{v_{min}}\right)^2\]. When Sedna is closest to the sun at perihelion, it possesses maximum speed and kinetic energy, while at aphelion, it carries the minimum. Calculations show Sedna's maximum kinetic energy is about 153.63 times its minimum, demonstrating the dramatic shift in energy as it speeds up and slows down on its long voyage around the sun.