Problem 29 - Physics

Question

What fraction of the power radiated by the sun is intercepted by the planet Mercury? The radius of Mercury is \(2.44 \times 10^{6} \mathrm{~m}\), and its mean distance from the sun is \(5.79 \times 10^{10} \mathrm{~m}\). Assume that the sun radiates uniformly in all directions.

Step-by-Step Solution

Answer

The fraction of the power intercepted by Mercury is approximately \(9.14 \times 10^{-5}\).

Step 1: Understanding the Problem

The problem asks to find the fraction of the sun's power intercepted by Mercury. This involves understanding the geometry of the interception: Mercury intercepts sunlight as if it were a circular disk facing the Sun.

Step 2: Surface Area of Radiation Sphere

The total power radiated by the Sun spreads out uniformly over a spherical surface area. The radius of this sphere is Mercury's distance from the Sun, given as \(5.79 \times 10^{10} \mathrm{~m}\). The surface area of the sphere can be calculated using the formula: \(4\pi r^2\), where \(r\) is the distance to Mercury.

Step 3: Calculating the Surface Area of Sphere

Calculate the surface area of the sphere with radius \(5.79 \times 10^{10} \mathrm{~m}\):\[4 \pi (5.79 \times 10^{10} )^2\].

Step 4: Area of Mercury's Intercepting Disk

Mercury intercepts the sunlight as a disk perpendicular to the Sun's rays. Using its radius, this circular area can be calculated as: \( \pi (2.44 \times 10^{6})^2\).

Step 5: Calculating the Disk Area of Mercury

Calculate the area of Mercury's intercepting disk:\[\pi (2.44 \times 10^{6})^2\].

Step 6: Fraction of Power Intercepted

The fraction of power intercepted by Mercury is the ratio of the area of Mercury's intercepting disk to the surface area of the radiation sphere, given by:\[ \text{Fraction} = \frac{\pi (2.44 \times 10^{6})^2}{4 \pi (5.79 \times 10^{10} )^2} \].

Step 7: Simplifying the Fraction

Since \(\pi\) appears in both the numerator and denominator, it cancels out. Simplify the fraction to:\[ \frac{(2.44 \times 10^{6})^2}{4 (5.79 \times 10^{10} )^2} \].

Step 8: Calculating the Final Fraction

Calculate the simplified fraction to get the fraction of power intercepted by Mercury. For precise calculation, use the formula derived:\[ \frac{(2.44^2)}{4 \times (5.79^2)} \times \left( \frac{10^6}{10^{10}} \right)^2\].

Key Concepts

Radiation

Radiation refers to the emission of energy from a source in the form of waves or particles. In the context of this exercise, it specifically refers to the energy radiated by the sun. The sun emits a tremendous amount of energy continuously, which travels through space and reaches the planets in our solar system.

When considering the sun's radiation, it's important to note that it happens uniformly in all directions. This means that the sun's energy spreads out over a vast area, reducing its intensity as it travels further from the source. This is because the same amount of energy must cover an increasingly large surface area as it moves away.

• The sun radiates energy uniformly in all directions.
• This energy travels in straight lines and spreads over a spherical surface.
• Planets intercept only a fraction of this energy due to their relative sizes and distances.

Understanding radiation in this way helps us grasp why planets situated farther from the sun, like Mercury, intercept only a fraction of its total energy output. This uniform radiation also forms the basis for determining the geometry of interception and the concept of surface area in our solar system.

Geometry of Interception

In the context of this exercise, the geometry of interception refers to how much of the sun's emitted radiation Mercury actually catches. This intercept is determined by Mercury acting as a circular disk facing the sun. When sunlight reaches Mercury, it hits it as if targeting a flat surface.

To visualize, imagine holding a plate directly in front of a light source. The plate represents Mercury, the light source represents the sun, and the area of the plate symbolizes the amount of radiation intercepted. This is an example of a simple geometric concept where a sphere's surface and a flat circle intersect.

• Mercury's interception is modeled as a circular disk.
• The intercepted area is a direct circle facing the sun.
• Sunlight intercepted depends on the planet’s size and distance from the sun.

This geometric approach allows for calculations on intercepted sunlight, helping astronomers and physicists to understand similar interactions across different celestial bodies. Recognizing this, you can see why only a small fraction of the sun's power reaches any given planet in our solar system.

Surface Area

Surface area plays a crucial role in understanding how much sunlight a planet like Mercury can intercept. The exercise specifically involves two main surface areas: the surface area of the sphere created by the sun's radiating energy and the area of Mercury's circular disk.

Firstly, consider the sun's energy as expanding evenly in a spherical shape as it travels outward. The surface area of this sphere is calculated using the formula \( 4\pi r^2 \) where \( r \) is the distance between the sun and Mercury. This formula is essential because it gives us an idea of how widely the sun's energy disperses.

• The spherical surface area increases with distance.
• As distance doubles, the surface area through which energy spreads increases by the square of the distance.

Next, the area of Mercury's intercepting disk is determined using \( \pi r^2 \) where \( r \) is Mercury's radius. This disk is the circle that directly intercepts sunlight. The fraction of intercepted sunlight is found by comparing these two areas, highlighting the significance of understanding surface areas in physics.

Through these calculations, we determine how a relatively small celestial body like Mercury interacts with a massive energy source like the sun. This concept underlines many principles of physics, such as energy distribution and geometric properties in space.