Problem 26 - University Physics with Modern Physics

Question

Planet Vulcan. Suppose that a planet were discovered between the sun and Mercury, with a circular orbit of radius equal to \(\frac{2}{3}\) of the average orbit radius of Mercury. What would be the orbital period of such a planet? (Such a planet was once postulated, in part to explain the precession of Mercury's orbit. It was even given the name Vulcan, although we now have no evidence that it actually exists. Mercury's precession has been explained by general relativity.)

Step-by-Step Solution

Answer

Vulcan's orbital period is approximately 48 days.

Step 1: Understanding Kepler's Third Law

Kepler's Third Law tells us that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, this is expressed as: \( T^2 \propto a^3 \), where \( T \) is the orbital period and \( a \) is the semi-major axis of the orbit.

Step 2: Expressing Vulcan's Orbit in Mercury's Terms

Let \( a_m \) be the average orbit radius of Mercury and \( a_v = \frac{2}{3}a_m \) be the orbit radius of Vulcan. We need to find the orbital period \( T_v \) for Vulcan.

Step 3: Using Mercury's Orbital Data

We know Mercury's orbital period \( T_m \) is approximately 88 days. From Kepler's Law, we have \( T_m^2 \propto a_m^3 \) or simply \( (88)^2 = a_m^3 \) after considering proportionality constants.

Step 4: Applying Kepler's Law to Vulcan

Using Kepler's Third Law for Vulcan, we write \( T_v^2 = k(a_v)^3 \). Since \( a_v = \frac{2}{3}a_m \), substitute to get \( T_v^2 = k\left(\frac{2}{3} a_m\right)^3 \).

Step 5: Eliminating Proportionality Constant

From Mercury's data, we have \( 88^2 = k(a_m)^3 \). Hence, \( k = \frac{88^2}{a_m^3} \). Substitute in the expression for \( T_v \) to get \( T_v^2 = \frac{88^2}{a_m^3} \left(\frac{2}{3} a_m \right)^3 \).

Step 6: Simplifying Vulcan's Period Expression

Simplify the expression: \( T_v^2 = 88^2 \left(\frac{8}{27}\right) \) after canceling \( a_m^3 \) with \( a_m^3 \) from the term. Thus, \( T_v = 88 \cdot \sqrt{\frac{8}{27}} \).

Step 7: Calculating the Orbital Period of Vulcan

Calculate \( T_v \): \( T_v = 88 \cdot \sqrt{0.2963} \approx 88 \cdot 0.544 \approx 47.9 \) days. Therefore, Vulcan's orbital period is approximately 48 days.

Key Concepts

Orbital Period

The orbital period refers to the time it takes for a planet or any celestial object to complete one full orbit around its central body, such as a star. It is a crucial aspect of understanding celestial mechanics and is highly influenced by the distance from the central body. Kepler's Third Law provides a direct relationship between the orbital period and the size of the orbit, which is indicated by the semi-major axis. This law states that the square of the orbital period (\(T^2\)) is proportional to the cube of the semi-major axis (\(a^3\)).
This means, mathematically, if you have a planet orbiting at a certain distance, you can estimate how long it takes to orbit based on how far it is. In the context of the exercise about the hypothesized planet Vulcan, the orbital period of this planet is significantly shorter than Mercury's due to its closer proximity to the sun.
Understanding orbital periods allows us to predict celestial events and understand time scales within solar systems.

• Orbital period is vital in determining the length of a "year" for planets.
• A shorter semi-major axis means a shorter orbital period.
• This principle is used by astronomers to study exoplanets and distant celestial bodies.

Semi-Major Axis

The semi-major axis is a fundamental concept in celestial mechanics and planetary science. It is essentially half of the major axis of an orbit, representing the longest diameter of an elliptical orbit. In a circular orbit, the semi-major axis is the radius of the circle.
In the exercise concerning planet Vulcan, its semi-major axis is two-thirds the semi-major axis of Mercury, making it closer to the sun. This small distance significantly affects Vulcan's orbital period, as described by Kepler's Third Law. Essentially, a smaller semi-major axis reduces the orbital path, and thus, the time required for one orbit.
Understanding the semi-major axis:

• It is a primary determinant of how far a celestial body is from its star.
• In our solar system, it affects both the orbital speed and period of planets.
• This axis is crucial in calculations for satellite launches and space missions.

Planetary Orbits

Planetary orbits are the paths celestial bodies follow around their stars. They can be elliptical or circular, with the sun (or other central body) at one of the focal points for elliptical orbits. Nearly all planets in our solar system follow elliptical orbits, which range slightly from circular.
Kepler's Laws provide a framework for understanding these orbits. The Third Law, in particular, helps us understand the relationship between a planet's distance from the sun and its orbital period. In the case of Vulcan, its hypothetical orbit between the sun and Mercury suggests a unique set of orbital characteristics, such as a shorter year due to its closer orbit.
Key points about planetary orbits:

• They are shaped by gravity and the initial velocity of the planet.
• The orbits determine seasons, climates, and year lengths on planets.
• Understanding orbits allows astronomers to track and predict movements of planets and asteroids.