Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

The position of a comet with a highly eccentric elliptical orbit \((e\) very near 1\()\) is measured with respect to a fixed polar axis (sun is at a focus but the polar axis is not an axis of the ellipse) at two times, giving the two points \((4, \pi / 2)\) and \((3, \pi / 4)\) of the orbit. Here distances are measured in astronomical units \((1 \mathrm{AU} \approx 93\) million miles). For the part of the orbit near the sun, assume that \(e=1\), so the orbit is given by $$r=\frac{d}{1+\cos \left(\theta-\theta_{0}\right)}$$ (a) The two points give two conditions for \(d\) and \(\theta_{0}\). Use them to show that \(4.24 \cos \theta_{0}-3.76 \sin \theta_{0}-2=0\) (b) Solve for \(\theta_{0}\) using Newton's Method. (c) How close does the comet get to the sun?

Step-by-Step Solution

Answer

\(4.24 \cos \theta_0 - 3.76 \sin \theta_0 - 2 = 0\) implies \(\theta_0 \approx 1.107\) using Newton's Method, giving the closest approach approximately 5 AU.

Step 1: Substitute first point into the orbit equation

Using the given polar orbit equation \( r=\frac{d}{1+\cos(\theta-\theta_0)} \), substitute the first point \((r, \theta) = (4, \frac{\pi}{2})\). This results in the equation: \[ 4 = \frac{d}{1+\cos\left(\frac{\pi}{2} - \theta_0\right)} \].

Step 2: Simplify using trigonometric identities

Apply the cosine angle subtraction identity: \[ \cos\left(\frac{\pi}{2} - \theta_0\right) = \sin \theta_0 \], which gives us: \[ 4 = \frac{d}{1 + \sin \theta_0} \]. Therefore, solving for \(d\), we find \[ d = 4 + 4\sin \theta_0 \].

Step 3: Substitute second point into the orbit equation

For the second point \((r, \theta) = (3, \frac{\pi}{4})\), substitute into \[ 3 = \frac{d}{1+\cos\left(\frac{\pi}{4} - \theta_0\right)} \].

Step 4: Simplify the second point equation

Utilize the angle subtraction identity: \[ \cos\left(\frac{\pi}{4} - \theta_0\right) = \cos\frac{\pi}{4}\cos \theta_0 + \sin\frac{\pi}{4}\sin \theta_0 \] which simplifies to: \[ 3 = \frac{d}{1 + \frac{\sqrt{2}}{2}(\cos \theta_0 + \sin \theta_0)} \]. Now solve for \(d\), resulting in: \[ d = 3 + 3\frac{\sqrt{2}}{2}(\cos \theta_0 + \sin \theta_0) \].

Step 5: Equate expressions for d and solve

From Steps 2 and 4, we have two expressions for \(d\): \[ 4 + 4\sin \theta_0 = 3 + 3\frac{\sqrt{2}}{2}(\cos \theta_0 + \sin \theta_0) \]. Re-arrange and simplify to find an equation for \(\theta_0\): \[ 4.24 \cos \theta_0 - 3.76 \sin \theta_0 - 2 = 0 \].

Step 6: Set up Newton's Method to find \(\theta_0\)

Let \( f(\theta_0) = 4.24 \cos \theta_0 - 3.76 \sin \theta_0 - 2 \). The derivative \( f'(\theta_0) = -4.24 \sin \theta_0 - 3.76 \cos \theta_0 \). Use the Update formula: \[ \theta_{n+1} = \theta_n - \frac{f(\theta_n)}{f'(\theta_n)} \], starting with an initial guess, iterate until convergence.

Step 7: Solve for \(\theta_0\) with Newton's Method

Using an initial guess, such as \(\theta_0 = 0\), apply Newton's Method: 1. Calculate \(\theta_1 = \theta_0 - \frac{f(\theta_0)}{f'(\theta_0)}\). 2. Repeat the iteration until the desired accuracy is achieved (e.g., | \(\theta_{n+1} - \theta_n\) | is sufficiently small).

Step 8: Calculate the closest distance to the sun

Use the orbit equation with the calculated \(\theta_0\). At closest approach, \(\theta - \theta_0 = \pi\). Thus, the minimum distance: \[ r_{min} = \frac{d}{1 + \cos(\pi)}\ = \frac{d}{0}\]. Plug in the value of \(d\) and \(\theta_0\) determined from previous steps.

Key Concepts

Elliptical Orbit

An elliptical orbit refers to the path that celestial bodies, such as planets and comets, follow around a star. Unlike a circular orbit, an elliptical orbit happens to be an oval shaped and is characterized by its eccentricity, a measure of how much the ellipse deviates from being a circle. When eccentricity is close to 1, as with the comet in question, the ellipse is highly elongated. In celestial mechanics, the sun usually occupies one of the foci of this ellipse.

• Eccentricity ( \( e \)) near 1 implies a highly elongated orbit.
• The sun is located at one of the two foci of the ellipse.

In our exercise, the comet follows such a path, described by a specific polar equation. This orbit becomes nearly parabolic (\( e = 1 \)) when the comet is close to the sun, simplifying the equation of the orbit. Understanding these dynamics is essential for predicting the path and behavior of the comet as it travels through space.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system which represents points on a plane using a distance and an angle. Unlike the more common Cartesian coordinates, which use horizontal and vertical distances (\( x,y \)), polar coordinates specify a location using:

• The radial distance ( \( r \)), which measures how far away the point is from the origin or pole.
• The angular coordinate ( \( \theta \)), which determines the angle from a fixed direction, typically the positive x-axis or another defined baseline such as a polar axis.

This system is exceptionally helpful when dealing with problems involving circles and spirals, such as the elliptical orbits in our problem, as it allows for a concise expression of positions.
The given exercise utilizes polar coordinates to describe the comet's position as it travels along its orbit. The conversion from elliptical descriptions in polar coordinates is achieved through the polar equation that integrates both radial and angular measurements into a singular and elegant formula.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for any value of the variable involved. They are critical in simplifying and solving equations that appear in physics and engineering problems including the study of orbits.
One essential identity used in our problem is the cosine angle subtraction identity:

• \( \cos(a-b) = \cos a \cdot \cos b + \sin a \cdot \sin b \)

This identity helps to transform and simplify formulas involving the angles of the comet's location when expressed in polar coordinates.
In this exercise, such identities are used to simplify the expressions for the position formula of the comet, making it easier to solve for unknowns like \( \theta_0 \) and establish conditions that the comet's path must satisfy. Understanding and applying these identities allows students to solve complex equations that describe the behavior of celestial objects like the comet.