Question

What is the incoming solar irradiance (S) at the top of the atmosphere averaged over the whole Earth and over one year (in \(\mathrm{W} / \mathrm{m}^{2}\) ?

Short answer

The average incoming solar irradiance at the top of the Earth's atmosphere is approximately 340.25 W/m².

Step-by-step solution

Understand the Earth-Sun Relationship

Incoming solar irradiance at the top of the atmosphere is also known as the solar constant, and its value is approximately 1361 W/m². This is the amount of solar energy received per unit area at a perpendicular surface at the top of the atmosphere. However, this value must be averaged over the entire surface of the Earth and over one year.

Determine the Earth's Cross-sectional Area

The Earth's cross-sectional area is a circle with radius equal to the Earth's radius. The formula for the area of a circle is \( ext{Area} = \pi imes r^2\), where \r\ is the radius of the Earth.

Calculate the Total Solar Energy Received

The total solar energy received by the Earth is given by the product of the solar constant (1361 W/m²) and the cross-sectional area of the Earth. This calculation gives us the total solar energy received by the Earth over one second.

Average Over Earth's Surface

The Earth is a sphere, and thus its total surface area is \( ext{Area}_{ ext{sphere}} = 4 \pi imes r^2\), where \r\ is the radius of the Earth. To average the total solar energy received over the surface area of the Earth, we divide the total solar energy received by the Earth's total surface area.

Calculate the Average Solar Irradiance

Using the formula for the solar constant \(S = \frac{L}{4 \pi r^2}\), where \L\ is the solar luminosity, and dividing by 4 accounts for averaging over day and night and all Earth's surfaces. The average solar irradiance becomes \S_{\text{average}} = \frac{1361 \ ext{W/m}^2}{4} = 340.25 \ ext{W/m}^2\.

Key concepts

Solar Constant

When we talk about the solar constant, we're referring to the amount of solar energy received per unit area by a surface perpendicular to the Sun's rays at the top of the Earth's atmosphere. The solar constant has a value of approximately 1361 watts per square meter (W/m²). This figure represents a baseline measurement of solar power that reaches Earth from the Sun.
To clarify further, the solar constant is not affected by the Earth's atmosphere or its rotation. It is an average measurement, helping scientists understand how much solar energy the Earth intercepts at any given moment. However, this energy does not hit the Earth's surface directly. It needs to be spread over the entire planet and averaged over time.

Earth-Sun Relationship

The Earth-Sun relationship is a critical aspect of understanding solar irradiance. This relationship explains how the Earth moves in relation to the Sun. The Earth orbits the Sun in an elliptical path, which means the distance between the Earth and the Sun changes throughout the year.
Despite these changes, the solar constant remains relatively stable. This is due to the immense distance between the Earth and the Sun, which minimizes the effects of their varying distances. It's important because the solar energy received on Earth is foundational for climate studies, ecosystem dynamics, and even the development of solar energy technology.

Average Solar Energy

Averaging the solar energy involves spreading the solar constant over the entire Earth's surface and accounting for variations over a year. Since the Earth rotates and has an axial tilt, different parts receive varying solar energies. Additionally, half of the Earth is always in darkness, reducing the average.
To find this average solar energy, we take the solar constant (1361 W/m²) and divide it by four. The factor of four arises because:

• The cross-section of the Earth is a circle, but it has a 3D surface area equivalent to four of those circular areas.
• Accounting for day-night cycles as only parts of Earth receive sunlight at a time.

This gives an average value of about 340.25 W/m².

Earth's Surface Area

The Earth's surface area is a crucial factor in solar energy distribution and involves complex calculations. Unlike its cross-sectional area (a simple circle), the Earth's surface is a sphere. The formula to find the surface area of a sphere is given by:\[ \text{Surface Area}_{\text{sphere}} = 4 \pi R^2 \]where \(R\) is the radius of the Earth. This accounts for the entire outer shell of the planet, comprising all fleshed parts exposed to the atmosphere and ultimately the incoming solar energy.
Making use of this understanding, scientists can better predict the warming effects of the Sun on different parts of the planet as well as model the potential energy available from solar sources anywhere on the surface.