Question

A comet in an extremely elliptical orbit about a star has, of course, a maximum orbit radius. By comparison, its minimum orbit radius may be nearly \(0 .\) Make plots of the potential energy and a plausible total energy \(E\) versus radius on the same set of axes. Identify the classical turning points on your plot.

Short answer

The plot of potential energy against radius falls steeply, showing that potential energy is inversely proportional to distance. The total energy remains nearly constant due to conservation of energy and varies only slightly with change in radius. The turning points, representing the maximum and minimum distances (apogee and perigee) the comet can be from the star, visibly occur where the two energy plots intersect.

Step-by-step solution

Expression for Potential Energy

The potential energy (PE) of the comet can be given by the equation: \(PE = - G M m / r\) where \(G\) is the gravitational constant, \(M\) is the mass of the star, \(m\) is the mass of the comet, and \(r\) is the radius of the comet's orbit. This equation tells us that potential energy is inversely proportional to \(r\), and will therefore be negative.

Expression for Total Energy

The total energy of the comet (E) can be given by the equation \(E = KE + PE\), where \( KE\) is the Kinetic Energy of the comet. For elliptical orbits, we know total energy is always negative. As PE reduces, KE increases to maintain the total energy constant i.e. \(E\) is less variable than \(PE\) with respect to \(r\).

Plotting the energies against radius

Plot the potential energy (PE) and total energy (E) against the radius (r). The PE curve is a falling curve since PE is inversely proportional to the radius. The E curve is nearly horizontal, indicating the conservation of total energy in orbits.

Identifying classical turning points

The classical turning points on the plot are where the PE curve intersects the E curve. These points represent the maximum and minimum distances the comet can be from the star, i.e. apogee and perigee, where the kinetic energy is at a minimum or maximum respectively.

Key concepts

Potential Energy in Orbits

Understanding potential energy in celestial orbits is crucial to grasping orbital mechanics. In simple terms, potential energy (PE) represents the energy stored within a comet's position relative to a star. According to the universal law of gravitation, the PE of an object in space is given by the equation
\( PE = - \frac{G M m}{r} \), where
- \(G\) is the gravitational constant,
- \(M\) is the mass of the star,
- \(m\) is the mass of the comet,
- \(r\) is the distance from the star to the comet.

It's negative because the reference point at which PE is zero is infinitely far from the star. As the comet approaches the star, \(r\) decreases, causing PE to become more negative, indicating that the system's energy is lower when the objects are close together. This negative sign conforms to the concept that work is required to separate the comet and the star against the gravitational pull—hence, it has higher PE when it is far away and lower PE when near the star.

For example, if a comet's orbit brings it closer to the star, its kinetic energy increases while its PE decreases, highlighting the trade-off between velocity and position in the conservation of energy.

Total Energy of Comets

Comets are renowned for their spectacular journeys around stars, which are governed by the laws of physics. The total energy \(E\) of a comet, which includes both kinetic energy (KE) and potential energy (PE), remains constant throughout its orbit given there's no external force acting on it. The total energy equation is \(E = KE + PE\).

For a comet in an elliptical orbit, this energy is always negative, indicating that the comet is bound to the star's gravitational field. The absolute value of this negative total energy is a measure of the comet's binding energy to the star. The relationship between KE and PE during the comet's orbit is such that when the comet is closest to the star (at perihelion), its velocity—and thus its KE—is at a peak while its PE is at its most negative. Conversely, at the farthest point (aphelion), the comet's velocity and KE are at their minimum, and the PE is less negative as \(r\) is larger.

The comet's constant total energy implies that its motion can be predicted based on the conservation of energy, which is a foundational principle in orbital mechanics. It's fascinating how the interplay between kinetic and potential energy shapes the comet's trajectory as it carves its path around the star.

Classical Turning Points

In the grand dance of celestial bodies, classical turning points act as the extremities of the comet's travel within its orbit around a star. These points correspond to the specific instants when the comet's direction of motion reverses. At these turning points, the comet's kinetic energy (KE) is either maximized or minimized.

On a graph of potential energy versus radius, these turning points are identifiable at the intersections where the curve of potential energy and the line of constant total energy meet. In a graphically represented elliptical orbit, these intersections mark the perihelion and aphelion—the comet's closest and farthest distances from its star, respectively.

At perihelion, the comet has its maximum kinetic energy and minimum potential energy, moving at its fastest pace. Conversely, at aphelion, its kinetic energy is at a minimum and potential energy is at a maximum in its negative value range, moving more sluggishly. These classical turning points are instrumental for understanding orbital periods and the longevity of a comet's orbit, providing astronomers and astrophysicists critical data for charting cosmic navigation and understanding dynamic gravitational systems.