1) Every second the Sun outputs 3.90* 10^26 Joules of energy. Power is energy output per unit
time and is given in units of Watts (W) where 1 W = 1 Joule per second. The Sun's power
output is therefore 3.90*10^26 Watts. This is called the solar luminosity. As this energy travels
outward from the Sun it is spread (fairly evenly) over a spherical surface that increases in area
with distance from the Sun (recall the area of a sphere is proportional to its radius squared).
The energy received each second by an area of 1 m2 is called the energy flux.
a) Calculate the energy °ux from the Sun at a distance of 9:54 AU (that is, Saturn's mean
distance) from the Sun. How does this compare with the solar flux received at the Earth's
mean distance from the Sun?
b) Suppose we place a perfectly black sphere with radius 2600 km at a distance of 9:54 AU from
the Sun. How much direct solar energy does the sphere absorb in one second?
c) If the sphere in part (b) is in thermal equilibrium, what is its temperature?
d) Titan re°ects approximately 20% of the light that strikes its surface, and absorbs the rest.
Titan's radius is roughly 2600 km. How much energy from sunlight does Titan absorb each
second? What is the temperature of Titan if it has no internal heat sources of significance?
These are exercises in using the inverse square law of luminosity and the blackbody law of thermal radiation. You will be helped if you first make an effort and show your work.
a) To calculate the energy flux from the Sun at a distance of 9.54 AU from the Sun, we need to consider the inverse square law. The energy flux is inversely proportional to the square of the distance from the source.
First, let's calculate the energy flux at Earth's mean distance from the Sun. The mean distance from the Sun to Earth is approximately 1 Astronomical Unit (AU), which is about 150 million kilometers.
Using the formula for energy flux, we have:
Energy Flux at Earth = Solar Luminosity / (4 * pi * (Earth-Sun distance)^2)
Energy Flux at Earth = 3.90 * 10^26 Watts / (4 * pi * (1 AU)^2)
Now, let's calculate the energy flux at a distance of 9.54 AU, which is Saturn's mean distance from the Sun:
Energy Flux at Saturn = Solar Luminosity / (4 * pi * (9.54 AU)^2)
To compare the two, we can calculate the ratio of the two energy flux values:
Ratio = Energy Flux at Saturn / Energy Flux at Earth
b) To calculate the energy absorbed by the black sphere at a distance of 9.54 AU from the Sun, we need to consider the energy flux and the surface area of the sphere.
The energy absorbed is given by:
Energy Absorbed = Energy Flux at 9.54 AU * Surface Area of the Sphere
Surface Area of the Sphere = 4 * pi * (Radius of the Sphere)^2
c) To calculate the temperature of the sphere in thermal equilibrium, we can use the Stefan-Boltzmann Law, which states that the power radiated by an object is proportional to its surface area and the fourth power of its temperature.
The power radiated by the sphere is equal to the power absorbed from the Sun:
Power Radiated = Power Absorbed
Using the Stefan-Boltzmann Law, we have:
Emissivity * Surface Area of the Sphere * (Temperature)^4 = Energy Absorbed
Solving for Temperature, we can find the temperature of the sphere.
d) To calculate the energy absorbed by Titan from sunlight, we need to consider the reflection and absorption percentages. Let's assume that the solar energy received at Titan is the same as the solar flux at Saturn's distance.
Energy Absorbed by Titan = (1 - Reflection Percentage) * Energy Flux at Saturn * Surface Area of Titan
Using the Stefan-Boltzmann Law, we can find the temperature of Titan by assuming it's in thermal equilibrium with the absorbed energy. Use the same equation as in part (c) to solve for the temperature.