Problem 17 - Classical Mechanics

Question

One classic there in science fiction is a twin plaret ("Planet \(X\) ) to Earth that is identical in mass, energy, and Mentholatum twit is located on the urtut \(90^{\circ}\) cut of phase with Earth so that it is lidulca froce the Sun. However, because of the elliptical nature of the critic it is not always completely hidden. Assume this twin planet is in the same Keplenan ortut as Earth in such a manner the is ts in apheling when Earth is in peribetion. Calculate to hist order in the egocentricity \(e\) the maximentn angular separation of the twin and the Sun as viewed from the Earth. Could such a twin be visible from Earth? Suppose the twin planet is in an elliptical orbit having the seem sice and shope as that of Earth, but rotated \(180^{\circ}\) from Earth's orbit, so that Earth and the twin are in rebellion at the same time. Repeat your calculation and compare the visibility in the two situations.

Step-by-Step Solution

Answer

The maximum angular separation is approximately \(0.0334\) radians, making the twin planet almost invisible from Earth.

Step 1: Understanding the Problem

We need to calculate the maximum angular separation between the twin planet and the Sun as viewed from Earth, given that the twin planet's orbit is identical to Earth's but rotated in phase. This involves understanding how maximum angular separation depends on orbital configuration and eccentricity.

Step 2: Review Key Concepts

The maximum angular separation in orbit depends on the positions of the planets at aphelion and perihelion, given Earth's elliptical orbit with eccentricity \(e\). At perihelion, Earth's distance is minimal, and its maximum distance occurs at aphelion.

Step 3: Consider Twin's Angular Position

The twin planet is at aphelion when Earth is at perihelion and vice versa. We need to compute the angular separation when both Earth and the twin are at these points in opposition, an angle contributed by their separation distance.

Step 4: Use Geometry of Elliptical Orbits

Using the positions of perihelion and aphelion, the semi-major axis \(a\) and eccentricity \(e\), the distance from the Sun to the planet can be described by \( r = a(1 - e)\) at perihelion and \( r = a(1 + e) \) at aphelion.

Step 5: Calculate Angular Separation

The angular separation \(\theta\) can be approximated using the formula: \[ \theta \approx 2e \]. This considers the angular positions at aphelion and perihelion are roughly \(\theta = 2\arctan\left(\frac{ae}{a}\right) \approx 2e\) for small \(e\).

Step 6: Evaluate Visibility

Given that \(\theta = 2e\) and Earth's orbit has \(e = 0.0167\), the angular separation is very small (about \(0.0334\) radians). Thus, the twin planet would not be visible to the naked eye due to its proximity to the Sun.

Step 7: Consider Second Configuration

In the second configuration where the orbit is rotated by \(180^{\circ}\), the twin and Earth reach perihelion simultaneously but on opposite points of the orbit. This results in the largest physical separation but near-zero angular separation as they are on opposite sides of the Sun.

Key Concepts

Elliptical Orbits

In orbital mechanics, an elliptical orbit is the path an object takes around a larger body due to gravity. Unlike a circular orbit, an elliptical orbit has an eccentric shape, defined by its eccentricity, a measure of how much the orbit deviates from being a perfect circle. This shape means there are points in the orbit where the orbiting body is closer or further from the object it orbits. This is influenced by two specific points known as the perihelion and aphelion. An elliptical orbit has two main characteristics: the semi-major axis and the eccentricity. The semi-major axis is the longest diameter of the ellipse, and the eccentricity describes how stretched the ellipse is. Most planets, including Earth, have orbits that are nearly circular but still slightly elliptical.
Understanding elliptical orbits helps us predict how celestial bodies move, explains variances in speed, and provides insights into gravitational forces at play. In such orbits, when a planet is closer to the body it's orbiting, it moves faster due to increased gravitational attraction.

Angular Separation

Angular separation refers to the visual distance between two objects in the sky, as observed from a specific point, such as Earth. It is measured in degrees, radians, or arcseconds and quantifies how far apart two celestial bodies appear to be. For example, the maximum angular separation between the twin planet and the Sun—as proposed in our scenario—can be computed based on their positions in the elliptical orbits.
This measurement is crucial in astronomy to determine how far apart celestial bodies appear, which helps astronomers track movements and predict alignments. In our problem, this concept aids in calculating the visible distance between the twin planet and the Sun from Earth when both are at opposite points in their elliptical orbits. A small angular separation implies that from the viewer's viewpoint, the objects appear close together, even if, physically, they are far apart. That's why at particular configurations, a celestial body might become invisible as it blends against the Sun's vast brightness.

Kepler's Laws

Kepler's laws of planetary motion are essential principles that describe how planets orbit around the Sun. They are foundational to understanding celestial mechanics. The laws are:

• The orbit of a planet is an ellipse, with the Sun at one of the two foci.
• A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, meaning a planet moves faster when it's closer to the Sun.
• The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

These laws help us understand the dynamics of the twin planet scenario by explaining how it shares the same orbital shape and size as Earth, thereby dictating similar angular velocities. Kepler's second law, in particular, clarifies why the angular separation varies as the planets approach perihelion and aphelion, leading to differing speeds and separations during their orbit.

Perihelion and Aphelion

Perihelion and aphelion are terms that describe the closest and farthest points, respectively, of a celestial object's elliptical orbit around the Sun. These terms are crucial in understanding the variation in Earth's orbit and the twin planet scenario.
At perihelion, the planet is closest to the Sun, experiencing stronger gravitational pull, leading it to move faster in its orbit. Conversely, at aphelion, the planet is farther away, the gravitational force is weaker, and the orbital speed reduces. Earth's perihelion happens in early January, whereas aphelion occurs in early July.
In the exercise scenario, when Earth is at perihelion, the twin planet is at aphelion, and vice versa. This alternating position influences the angular separation between the two as seen from Earth. Because of these configurations, despite physical closeness during opposition, the angular separation might remain small, explaining why the twin might not be visible from Earth.