Solar constant
Measure the solar constant using a simplified pyrometer connected to FizziQ Connect and estimate the extraterrestrial irradiance by Bouguer's method.
The solar constant, or total solar irradiance, is the power received per square meter by a surface perpendicular to the Sun's rays at the top of the atmosphere. Its value, approximately 1370 W/m², is fundamental to climate science and solar energy engineering. By measuring the heating rate of a blackened metal cylinder exposed to sunlight and applying Bouguer's method to correct for atmospheric absorption, you can estimate this constant from the ground.
Activity overview:
The student measures the solar irradiance at ground level using a pyrometer and extrapolates to the top of the atmosphere using Bouguer's method.
Level:
High school
FizziQ
Author:
Duration (minutes) :
120
What students will do :
- Measure solar irradiance at ground level with a simplified pyrometer
- Apply Bouguer's method to extrapolate irradiance to the top of the atmosphere
- Understand atmospheric absorption of solar radiation
- Calculate the Sun's surface temperature from the solar constant
- Use thermal measurements for energy calculations
Scientific concepts:
- Solar constant (~1370 W/m²)
- Atmospheric absorption
- Bouguer's method
- Stefan-Boltzmann law
- Thermal equilibrium
- Zenith angle and air mass
Sensors:
- Temperature probe connected to FizziQ Connect
Material needed:
- Smartphone or tablet with FizziQ Connect
- M5 Stack with temperature probe
- Blackened metal cylinder pyrometer
- Insulated housing
- Protractor and plumb line
- Stopwatch
Experimental procedure:
Weigh the metal cylinder and note its mass m (in kg). Note its specific heat capacity c (in J/(kg·K)), given by the manufacturer.
Measure the radius Rc of the cylinder's circular cross-section to calculate the receiving area S = π × Rc².
Connect the temperature probe to the M5 Stack and open FizziQ Connect. Place the probe in contact with the metal cylinder.
Set up the pyrometer with the shutter closed, orienting its entrance perpendicular to the Sun. Use the shadow cast by the device to align.
Wait for thermal equilibrium (stable temperature for 1 minute). Start recording and note the initial temperature θi.
Open the shutter at time ti. The cylinder heats up in the sunlight. Record for 2 to 5 minutes.
Close the shutter and note the final temperature θf at time tf. Calculate Δt = tf - ti and Δθ = θf - θi.
Measure the Sun's zenith angle z using the protractor and plumb line (z = 90° - Sun's altitude above the horizon), or calculate it from time and date.
Calculate the ground-level irradiance: Es = m × c × Δθ / (S × Δt). Repeat measurements at different times of day (4 to 6 measurements).
Plot log(Es) versus 1/cos(z). The y-intercept of the straight line gives log(E_HA), where E_HA is the extraterrestrial irradiance. Compare with 1370 W/m².
Expected results:
Ground-level irradiance Es typically varies from 400 to 900 W/m² depending on the Sun's height, sky clarity, and time of day. The plot of log(Es) versus 1/cos(z) gives a straight line whose y-intercept extrapolates to E_HA ≈ 1200-1500 W/m². With careful measurements, the value is within 10-20% of the accepted 1370 W/m².
Scientific questions:
- Why does ground-level irradiance vary during the day?
- What gases in the atmosphere are responsible for absorbing solar radiation?
- Why does Bouguer's method use the logarithm of irradiance?
- How can you estimate the Sun's surface temperature from the solar constant?
- What is the difference between direct and diffuse solar radiation?
- How is the solar constant used in climate models?
Scientific explanations:
The solar constant E₀ ≈ 1370 W/m² is the radiative power received by a 1 m² surface perpendicular to the Sun's rays at the mean Earth-Sun distance (1 AU), outside the atmosphere.
Ground-level irradiance is lower than the solar constant because the atmosphere absorbs and scatters part of the radiation. Bouguer's method corrects for this absorption by measuring at multiple zenith angles.
The atmospheric path length depends on the zenith angle z: L = h/cos(z), where h is the effective atmosphere thickness. At the zenith (z = 0), the path is shortest; near the horizon, it is longest.
By plotting log(E) versus 1/cos(z), one obtains a straight line whose y-intercept gives log(E_HA), the extraterrestrial irradiance. The slope gives the atmospheric extinction coefficient.
The power received by the cylinder is P = m × c × Δθ / Δt, where m is the cylinder mass, c its specific heat capacity, and Δθ/Δt the heating rate. Dividing by the receiving area S gives the irradiance.
To estimate the Sun's surface temperature, one uses the Stefan-Boltzmann law: Φ = σ × T⁴. Combining the solar constant with the Sun-Earth distance and the Sun's radius gives T ≈ 5800 K.
Extension activities:
- Why does ground-level irradiance vary during the day?
- What gases in the atmosphere are responsible for absorbing solar radiation?
- Why does Bouguer's method use the logarithm of irradiance?
- How can you estimate the Sun's surface temperature from the solar constant?
- What is the difference between direct and diffuse solar radiation?
- How is the solar constant used in climate models?
Frequently asked questions:
Q: My extrapolated value is far from 1370 W/m².
R: The method is sensitive to atmospheric conditions. Haze, thin clouds, and pollution increase extinction. Perform measurements on a very clear day.
Q: I only have time for 2-3 measurements. Is that enough?
R: Bouguer's method requires at least 4 measurements at different solar heights to give a reliable extrapolation. Fewer points make the line fit unreliable.
Q: How do I align the pyrometer with the Sun?
R: Point the entrance directly at the Sun so that its shadow is minimal (a circular opening casts a circular shadow). Adjust until the shadow is smallest.
Q: Can I do this experiment without a pyrometer?
R: You can use a blackened can with a temperature probe inside, but precision will be lower. The key is knowing the mass, specific heat, and receiving area accurately.