Problem 62 - A Complete Resource Book in Mathematics for JEE Main

Question

The orbit of the earth is an ellipse with eccentricity \(\frac{1}{60}\) with the sum at one focus, the major axis being approximately \(186 \times 10^{6}\) miles in length. The shortest and longest distance of the earth from the sun is (A) \(9145 \times 10^{4}\) miles, \(9455 \times 10^{4}\) miles (B) \(9147 \times 10^{4}\) miles, \(9457 \times 10^{4}\) miles (C) \(9145 \times 10^{6}\) miles, \(9455 \times 10^{6}\) miles (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C): the shortest distance is 9145 × 10⁶ miles, and the longest is 9455 × 10⁶ miles.

Step 1: Understand the Problem

We need to find the shortest and longest distances (perihelion and aphelion) between the Earth and the Sun, given the ellipse's eccentricity and major axis length.

Step 2: Relate the Given Values to Elliptical Parameters

The major axis length of the ellipse is given as \(2a = 186 \times 10^6\) miles, so the semi-major axis \(a = \frac{186 \times 10^6}{2} = 93 \times 10^6\) miles. The eccentricity \(e\) is given as \(\frac{1}{60}\).

Step 3: Calculate the Linear Eccentricity

The linear eccentricity \(c\) is calculated as \(c = ae\). Substituting the given values, \(c = 93 \times 10^6 \times \frac{1}{60} = 1.55 \times 10^6\) miles.

Step 4: Find the Shortest Distance (Perihelion)

The perihelion distance (shortest distance) \(r_{min}\) is given by \(a - c\). Therefore, \(r_{min} = 93 \times 10^6 - 1.55 \times 10^6 = 91.45 \times 10^6\) miles.

Step 5: Find the Longest Distance (Aphelion)

The aphelion distance (longest distance) \(r_{max}\) is given by \(a + c\). Thus, \(r_{max} = 93 \times 10^6 + 1.55 \times 10^6 = 94.55 \times 10^6\) miles.

Step 6: Determine the Correct Answer Choice

Compare the calculated shortest and longest distances with the given choices. The shortest distance of \(91.45 \times 10^6\) miles and longest distance of \(94.55 \times 10^6\) miles match answer choice (C): \(9145 \times 10^6\) miles, \(9455 \times 10^6\) miles.

Key Concepts

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a perfect circle. In mathematics, the eccentricity \(e\) of an ellipse is calculated using the formula \(e = \frac{c}{a}\), where \(c\) is the distance from the center of the ellipse to a focus, and \(a\) is the semi-major axis length. A circle, having no deviation, has an eccentricity of 0. As the eccentricity approaches 1, the ellipse becomes more stretched out and resembles a line segment. For celestial bodies, such as planets orbiting stars, the eccentricity is often very small, making the orbits nearly circular but still distinctly elliptical. In our example with Earth's orbit around the Sun, the eccentricity is given as \(\frac{1}{60}\), indicating a nearly circular but slightly elliptical orbit. Understanding eccentricity helps in determining how the shape of Earth's path affects seasonal variations and the lengths of seasons.

Major Axis

The major axis of an ellipse is the longest diameter, stretching from one end of the ellipse through its foci to the other end. It divides the ellipse into two equal halves. The semi-major axis, represented by \(a\), is half the length of the major axis and runs from the center of the ellipse to one end of the longest diameter.In the context of planetary orbits, the length of the semi-major axis is a crucial parameter because it helps define the size of the orbit and affects the orbiting body's orbital period. For Earth's orbit mentioned in our exercise, the major axis length is about \(186 \times 10^6\) miles, so the semi-major axis is half of this, or \(93 \times 10^6\) miles. This value is important not just for geometric reasons but also in understanding the time taken for Earth to complete one full orbit around the Sun, as described by Kepler's Third Law of Planetary Motion.

Perihelion and Aphelion

Perihelion and aphelion refer to the points in Earth's orbit where it is closest and farthest from the Sun, respectively. These points are of utmost importance in astronomy and affect seasonal weather patterns due to the variation in distance from the Sun.The perihelion, derived from "peri" meaning near and "helios" meaning sun, occurs when Earth is closest to the Sun. This distance \(r_{min}\) is calculated by the formula \(a - c\), where \(a\) is the semi-major axis and \(c\) is the linear eccentricity. In the given problem, \(r_{min} = 91.45 \times 10^6\) miles.On the other hand, aphelion, constructed from "apo" meaning away, is when Earth is farthest from the Sun. This distance \(r_{max}\) is given by the formula \(a + c\). For Earth's orbit, it is \(94.55 \times 10^6\) miles.Understanding these distances is critical, as even small changes in these values can lead to significant differences in solar energy received by Earth, influencing climate and environmental conditions.