Question

The motion of a planet in a circular orbit about a star obeys the equations of simple harmonic motion. If the orbit is observed edge-on, so the planet's motion appears to be one-dimensional, the analogy is quite direct: The motion of the planet looks just like the motion of an object on a spring. a) Use Kepler's Third Law of planetary motion to determine the "spring constant" for a planet in circular orbit around a star with period \(T\). b) When the planet is at the extremes of its motion observed edge-on, the analogous "spring" is extended to its largest displacement. Using the "spring" analogy, determine the orbital velocity of the planet.

Short answer

b) What is the expression for the orbital velocity of the planet when the "spring" is at its maximum displacement? a) The expression for the spring constant for a planet in a circular orbit around a star using Kepler's Third Law is: $$k = -\frac{GMm}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}x}$$ b) The expression for the orbital velocity of the planet when the "spring" is at its maximum displacement is: $$v = \sqrt{-\frac{GMx}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}}}$$

Step-by-step solution

State Kepler's Third Law

Kepler's Third Law relates the period of a planet's orbit (T) to the semi-major axis of its orbit (a) and the mass of the central star (M). In this problem, we can consider the planet to be moving around the star in a circular orbit. The Law states that: $$\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$$ where G is the gravitational constant.

Rewrite the equation in terms of the orbital period

We are given the period T and we need to find the "spring constant". We can rewrite the equation in step 1 in terms of T: $$a = (\frac{GMT^2}{4\pi^2})^{\frac{1}{3}}$$

Apply Hooke's Law and gravitational force formula

Hooke's law is given by: $$F = -kx$$ where F is the force exerted by the "spring", k is the spring constant, and x is the displacement from the equilibrium position. The gravitational force between the planet of mass m and the star of mass M is given by: $$F_g = \frac{GMm}{a^2}$$ At the extremes of the motion observed edge-on, the force exerted by the "spring" is equal to the gravitational force, so we can equate them: $$-kx = \frac{GMm}{a^2}$$ Now, we need to find the spring constant k.

Isolate the spring constant

To isolate the spring constant k, we can rewrite the previous equation: $$k = -\frac{GMm}{a^2x}$$ Substitute the expression for a from step 2: $$k = -\frac{GMm}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}x}$$ This expression gives the spring constant k in terms of the given period T. #a)

State the expression for the spring constant

Using Kepler's Third Law of planetary motion, the spring constant for a planet in circular orbit around a star with period T is: $$k = -\frac{GMm}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}x}$$ #b)

Determine the orbital velocity using energy conservation

The total mechanical energy of the planet in the orbit is the sum of its kinetic and potential energy. When the "spring" is at its maximum displacement, all the potential energy is converted into kinetic energy. Therefore: $$\frac{1}{2}mv^2 = \frac{1}{2}kx^2$$ Substitute the expression for k from step 5: $$\frac{1}{2}mv^2 = \frac{1}{2}(-\frac{GMm}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}x})x^2$$

Isolate the orbital velocity

Now, we need to find the orbital velocity v. We can simplify the expression and isolate v on one side of the equation: $$v^2 = -\frac{GMx}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}x}$$ Take the square root of both sides: $$v = \sqrt{-\frac{GMx}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}}}$$ This expression gives the orbital velocity of the planet when the "spring" is at its maximum displacement.

State the orbital velocity expression

Using the "spring" analogy, the orbital velocity of the planet when it is at the extremes of its motion observed edge-on is given by: $$v = \sqrt{-\frac{GMx}{(\frac{GMT^2}{4\pi^2})^{\frac{2}{3}}}}$$

Key concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion that is particularly important in physics and astronomy. It describes the motion of objects that experience a restoring force directly proportional to the displacement from an equilibrium position. This force can be visualized as the kind of tug a spring exerts on an object attached to it when it's stretched or compressed.

Mathematically, Hooke's Law captures this idea with the equation \[ F = -kx \], where \( F \) is the restoring force, \( k \) is the spring constant representing how stiff the spring is, and \( x \) is the displacement from equilibrium. The negative sign indicates that the force is always directed opposite to the displacement, pulling the object back towards the equilibrium position.

Perhaps unexpectedly, SHM can also describe orbital motion when observed from a certain point of view. In the case of a planet orbiting a star, if viewed edge-on, the motion mimics that of an object oscillating on a spring; the star’s gravitational pull playing the role of the restoring force. Understanding SHM helps us link gravitational force and orbital movement, unlocking a deeper comprehension of planetary dynamics.

Gravitational Force

Gravitational force is the attractive force that exists between any two objects with mass. According to Newton's Law of Universal Gravitation, the force between two objects can be calculated by the equation \[ F_g = \frac{GMm}{r^2} \], where \( G \) is the gravitational constant, \( M \) is the mass of one object (e.g., a star), \( m \) is the mass of the second object (e.g., a planet), and \( r \) is the distance between the centers of the two objects.

For planets in orbit around a star, this gravitational force provides the centripetal force necessary to keep them moving in a circular path. If used in the context of SHM, and combined with Hooke's Law, we can find a fascinating parallel that allows us to calculate an equivalent 'spring constant' for the orbit. This analogy emphasizes the universality of physical laws, showing how gravity can effectively act like a spring, keeping planets in a stable orbit around their stars.

Orbital Velocity

Orbital velocity refers to the speed at which an object moves around another object in orbit. For planets orbiting stars, this velocity depends on the distance from the star and the gravitational pull the star exerts on the planet.

Using our 'spring' analogy in the context of circular orbits, we can determine the orbital velocity of a planet at the point where the gravitational pull (analogous to the spring force at maximum displacement) is at its greatest. This is akin to looking at the planet's velocity when it is at the extremes of ellipse-like motion observed edge-on.

The conservation of energy principle allows us to determine this orbital velocity by equating kinetic energy with the potential energy stored in the 'spring'. Thus, \[ v = \sqrt{\frac{GM}{a}} \], where \( v \) is the orbital velocity, \( G \) is the gravitational constant, \( M \) is the mass of the star, and \( a \) is the semi-major axis of the orbit, which in our analogy translates to the maximum displacement of the 'spring'. Computing the orbital velocity using this relation provides critical information for understanding how celestial bodies move and interact through space, grounding the seemingly abstract concept of SHM in the concrete and observable movements of the heavens.