Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Mars's Elliptical Orbit. Mars's distance from the Sun varies from \(1.38 \mathrm{AU}\) to \(1.66 \mathrm{AU}\). How much does this change the globally averaged strength of sunlight over the course of the martian year? Give your answer as a percentage by which sunlight at perihelion (the orbital point closest to the Sun) is stronger than that at aphelion (the farthest orbital point). Comment on how this affects the martian seasons. (Hint: Remember that light follows an inverse square law; see Figure \(7.2 .\)

Short answer

The sunlight at Mars' perihelion is 44.7% stronger than at aphelion. This affects Martian seasons with more extreme temperature swings.

Step-by-step solution

Understanding the Inverse Square Law

The intensity of sunlight is inversely proportional to the square of the distance from the sun. This means if the distance between Mars and the Sun increases, the sunlight received decreases, and vice versa. Mathematically, this relationship is given by:\[ I \propto \frac{1}{d^2} \] where \( I \) is the intensity of sunlight, and \( d \) is the distance from the Sun.

Calculate the Intensity at Perihelion

At perihelion, the distance from the Sun is the minimum, \( d_1 = 1.38 \text{ AU} \). Thus, the intensity of sunlight received can be expressed as:\[ I_1 = \frac{1}{(1.38)^2} \].

Calculate the Intensity at Aphelion

At aphelion, the distance from the Sun is maximum, \( d_2 = 1.66 \text{ AU} \). The intensity of sunlight received at this point can be calculated using the same inverse square law:\[ I_2 = \frac{1}{(1.66)^2} \].

Find the Percentage increase

To find how much stronger sunlight is at perihelion than at aphelion, calculate the percentage increase in intensity using:\[ \text{Percentage Increase} = \left(\frac{I_1 - I_2}{I_2}\right) \times 100\%\]Substitute \( I_1 \) and \( I_2 \) from Steps 2 and 3,\[ \frac{\frac{1}{(1.38)^2} - \frac{1}{(1.66)^2}}{\frac{1}{(1.66)^2}} \times 100\% = \frac{1.66^2 - 1.38^2}{1.38^2} \times 100\% \]

Perform the Calculation

Calculate each term:\[ 1.66^2 = 2.7556 \]\[ 1.38^2 = 1.9044 \]\[ \text{Numerator} = 2.7556 - 1.9044 = 0.8512 \]\[ \text{Denominator} = 1.38^2 = 1.9044 \]\[ \text{Percentage Increase} = \frac{0.8512}{1.9044} \times 100\% \approx 44.7\% \].

Conclusion on Martian Seasons

The sunlight is about 44.7% stronger at perihelion than at aphelion. This significant change in solar intensity affects the Martian climate and seasons, making the temperature variation more extreme compared to Earth's more moderate seasonal changes.

Key concepts

Inverse Square Law

The inverse square law is a fundamental principle that describes how certain physical quantities, such as light intensity, diminish with distance. For planets like Mars, this law is crucial to understanding how the amount of sunlight it receives changes as it orbits the Sun.The inverse square law tells us that the intensity of sunlight is inversely proportional to the square of the distance between Mars and the Sun. This means that if Mars moves further away from the Sun, the sunlight intensity decreases by the square of that distance change. Mathematically, this relationship is expressed as:\[ I \propto \frac{1}{d^2} \]where \( I \) represents the intensity of sunlight, and \( d \) is the distance from the Sun. This principle is essential for calculating how solar intensity varies with Mars’s elliptical orbit, ranging between 1.38 AU and 1.66 AU. Just as a light source appears dimmer when you move away from it, the sunlight on Mars at aphelion (farthest point) is weaker compared to its strength at perihelion (closest point). This change influences important climatic processes on Mars.

Perihelion and Aphelion

Mars’s orbit around the Sun is elliptical, meaning it differs from a perfect circle. At perihelion, Mars is at its closest point to the Sun, about 1.38 AU away. Conversely, at aphelion, it is furthest from the Sun, about 1.66 AU away. These distances significantly affect the intensity of sunlight Mars receives.

• At perihelion, Mars receives the most intense sunlight of its orbit. This is because it is at the minimum distance from the Sun, resulting in more solar energy reaching the planet’s surface.
• At aphelion, the sunlight is less intense. Mars, being further away, receives less solar energy due to the increased distance.

The difference in sunlight intensity due to these distances is substantial. Using the inverse square law, we find that sunlight at perihelion is around 44.7% stronger than at aphelion. This significant difference is one reason for the extreme climate variations on Mars.

Martian Seasons

Seasons on Mars are greatly influenced by its elliptical orbit and the varying distance from the Sun throughout the Martian year. Unlike Earth, where seasons are primarily caused by the axial tilt, the variation in Mars's solar distance plays a larger role. During perihelion, Mars experiences its most intense solar radiation. This is like having a stronger "summer" even if the planet's axial tilt points away, leading to warmer temperatures and rapid changes in surface conditions. During aphelion, the opposite occurs, contributing to Mars's colder and longer "winters."

• The difference in solar energy leads to more dramatic temperature shifts compared to Earth’s relatively moderate seasonal variations. This can cause effects like higher wind speeds and more significant dust storms on Mars.
• Understanding these dynamics helps scientists predict Martian weather patterns and prepare for future manned missions.

These seasonal changes are critical for planning exploration and understanding Mars's past and present climate, which may offer insights into its potential for supporting life.