Problem 2 - Classical Mechanics - An Undergraduate Text

Question

A particle \(P\) of mass \(m\) moves under the attractive inverse square field \(\boldsymbol{F}=-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\). Initially \(P\) is at a point \(C\), a distance \(c\) from \(O\), when it is projected with speed \((\gamma / c)^{1 / 2}\) in a direction making an acute angle \(\alpha\) with the line \(O C\). Find the apsidal distances in the resulting orbit. Given that the orbit is an ellipse with \(O\) at a focus, find the semi-major and semi-minor axes of this ellipse.

Step-by-Step Solution

Answer

The apsidal distances and the semi-major and semi-minor axes of the ellipse depend on the eccentricity which is determined by the initial conditions of the particle's motion. These can be found by solving the equation for the ellipse and using the properties of the ellipse.

Step 1: Determine Particle Trajectory

In this type of motion, we know from basic mechanics that the trajectory of the particle is an ellipse and the point O is at one of the foci. Hence, the equation of the ellipse in polar coordinates (r, \(\theta\)) is \[r = \frac{l^2 / mγ}{1 + e cos(\theta)}\] where l is the magnitude of the angular momentum per unit mass and e is the eccentricity of the ellipse.

Step 2: Calculate Angular Momentum

The initial angular momentum per unit mass \(l\) is given by \(l = c(\gamma / c)^{1/2} sin(\alpha)\). Elliptical orbits are represented by eccentricities \(0 < e < 1\). If we substitute \(r = c\) and \(\theta = 180° - \alpha\) when the particle is at point C into the equation of the ellipse, we can solve for e. The value of \(l\) from above is used.

Step 3: Calculate Apsidal Distances

The apsidal distances or the distances from the foci to the points on the ellipse along the line of apses (the line through the major axis) are where \(r\) is the smallest and largest, and occur where \(d / d \theta [1 + e cos(\theta)] = 0\). Solving this gives \(\theta = 0\) and \(\theta = 180°\) respectively. Substituting these values into the equation of the ellipse gives the apsidal distances.

Step 4: Calculate Axes of the Ellipse

For an ellipse, \(r_1 + r_2 = 2a\), where a is the semi-major axis, and \(r_1 - r_2 = 2c\), where c is the distance from the focus to the center of the ellipse. With the apsidal distances computed in the previous step, we can solve for a and b. The semi-minor axis b can be found using the formula \(b = a \sqrt{1 - e^2}\) where e is the eccentricity.

Key Concepts

Inverse Square Law

The inverse square law is a fundamental principle in physics that governs the behavior of forces, such as gravity and electromagnetism. In this problem, the particle moves under the influence of an attractive force that follows this law:

• The force is proportional to the product of the masses involved.
• It is inversely proportional to the square of the distance between the two bodies.

This means that if the distance between two masses doubles, the force will reduce to a quarter of its original strength. In our exercise, the force experienced by the particle is given by:\[\boldsymbol{F} = -\left(\frac{m \gamma}{r^2}\right) \hat{\boldsymbol{r}}\]This illustrates how the force depends on the distance, demonstrating the concept of the inverse square nature of the gravitational force between two masses.

Elliptical Orbits

Elliptical orbits are a beautiful realization of celestial mechanics, explained by Kepler's Laws. In our context, the particle describes an elliptical trajectory under the inverse square law of attraction, with the central body located at one of the ellipse's foci. Key characteristics include:

• The semi-major axis, denoted by \(a\), defines the longest diameter of the ellipse.
• The semi-minor axis, \(b\), is the shortest diameter and is perpendicular to the semi-major axis.
• The eccentricity \(e\) determines the shape of the ellipse, where \(0 \le e < 1\).

The trajectory of the particle in polar coordinates can be described by the equation:\[r = \frac{l^2/m\gamma}{1 + e \cos(\theta)}\]This represents how distance from the focal point varies with the angle \(\theta\). Understanding elliptical orbits is crucial for deciphering the motion of celestial bodies like planets and moons.

Angular Momentum

Angular momentum is a conserved quantity in orbital mechanics, meaning it remains constant if there is no external torque. It is dependent on:

• Mass of the object
• Its velocity perpendicular to the direction of force
• The distance from the center of rotation

In the problem, angular momentum per unit mass \(l\) is calculated as:\[l = c \left(\frac{\gamma}{c}\right)^{1/2} \sin(\alpha)\]Here, \(c\) is the initial distance, \(\gamma\) relates to the gravitational parameter, and \(\alpha\) is the launch angle. This formula provides insight into how the speed and direction of motion contribute to maintaining the orbit. Without this conserved rotational motion, the shape and stability of the orbit would be affected.

Apsidal Distances

Apsidal distances are the extremities of an orbit's radius, specifically the closest (periapsis) and farthest (apoapsis) points from the central focus. Calculating these points allows us to understand the motion extent of orbiting bodies. From the ellipse equation:\[r = \frac{l^2/m\gamma}{1 + e \cos(\theta)}\]We find apsidal distances by determining where the derivative with respect to \(\theta\) is zero. This occurs at:

• \(\theta = 0\): Closest approach, periapsis.
• \(\theta = 180^\circ\): Farthest point, apoapsis.

By substituting these angles back into the equation, the respective distances can be determined. Understanding apsidal distances is essential in calculating orbital parameters like semi-major and semi-minor axes and in predicting orbital dynamics and durations.