The luminosity of the Sun is 4*10^33 erg/s, and its radius is 7*10^10 cm.
What is the Sun's effective temperature? Please enter your answer in units of Kelvin.
What is the Sun's effective temperature? Please enter your answer in units of Kelvin. - unanswered
What is the flux of sunlight on the Earth?
- unanswered
You are tasked with building a solar power plant in the Arizona desert, using solar panels with 10% efficiency. How large an area (km^2) must your solar panels cover to match the power output of a large nuclear powerplant (about a GigaWatt)? Please enter your answer in units of km^2
You are tasked with building a solar power plant in the Arizona desert, using solar panels with 10% efficiency. How large an area (km^2) must your solar panels cover to match the power output of a large nuclear powerplant (about a GigaWatt)? Please enter your answer in units of km^2 - unanswered
The Keck telescope on Mauna Kea has an angular resolution on Earth of half an arcsecond.
>>How far away (in meters) could you read ("resolve the letters of") a book with 3 mm square type, using the Keck telescope on Earth?
- unanswered
In space, the angular resolution of the Keck telescope is govererned by the diffraction limit.
How far away could you read the same book, using the Keck telescope in space? Please express your answers in units of meters.
To calculate the Sun's effective temperature, we can use the Stefan-Boltzmann law, which states that the luminosity of a star is related to its temperature and radius. The formula is as follows:
L = 4πR^2σT^4
Where:
L is the luminosity of the Sun (4*10^33 erg/s)
R is the radius of the Sun (7*10^10 cm)
σ is the Stefan-Boltzmann constant (5.67 x 10^-5 erg/(cm^2·s·K^4))
T is the effective temperature of the Sun (to be determined)
Rearranging the formula to solve for T, we have:
T^4 = L / (4πR^2σ)
T = (L / (4πR^2σ))^0.25
Now we can substitute the given values into the formula to find T:
T = (4*10^33 erg/s / (4π(7*10^10 cm)^2(5.67 x 10^-5 erg/(cm^2·s·K^4))))^0.25
Evaluating the expression, we find that the Sun's effective temperature is approximately 5778 Kelvin.
Moving on to the flux of sunlight on Earth, we can use the inverse square law. This law states that the intensity of radiation decreases as the square of the distance from the source increases. The formula is as follows:
F = L / (4πd^2)
Where:
F is the flux of sunlight on Earth (to be determined)
L is the luminosity of the Sun (4*10^33 erg/s)
d is the average distance from the Sun to Earth (1.496 x 10^13 cm)
Substituting the values into the formula, we can calculate the flux of sunlight on Earth:
F = (4*10^33 erg/s) / (4π(1.496 x 10^13 cm)^2)
Now, to calculate the area required for the solar power plant, we need to find the power output of a large nuclear power plant in terms of Watts. 1 GigaWatt is equivalent to 10^9 Watts. Since the solar panels have an efficiency of 10%, we need to divide the power output by the efficiency to find the incident power:
Power incident on the solar panels = (1 GigaWatt) / 0.1
Using this value, we can calculate the total power absorbed by the solar panels. Assuming the power incident on the panels is the same as the power absorbed:
Power absorbed by solar panels = Power incident on the solar panels
To calculate the area required, we can use the formula:
Area = Power absorbed by solar panels / (Luminosity of the Sun * Efficiency of solar panels)
Now, let's plug in the values:
Area = ((1 GigaWatt) / 0.1) / (4*10^33 erg/s * 0.1)
Simplifying the expression, we find that the solar panels must cover an area of approximately 2.5 km^2.
Moving on to the angular resolution of the Keck telescope on Earth, we have half an arcsecond. To determine how far away we can read a book with 3 mm square type, we can use the formula for angular resolution:
θ = λ / D
Where:
θ is the angular resolution (in radians)
λ is the wavelength of light (approximated to 500 nm or 5 x 10^-7 m)
D is the diameter of the telescope (to be determined)
Rearranging the formula to solve for D, we have:
D = λ / θ
Plugging in the values, we find that D = (5 x 10^-7 m) / (0.5 arcsecond).
Finally, to determine the distance at which we can read the same book using the Keck telescope in space, we need to consider the diffraction limit. The formula for the angular resolution due to diffraction is:
θ = 1.22 * (λ / D)
Using the same values as before, we can calculate θ, which will be the same as the angular resolution in space. With the angular resolution known, we can use the equation for the angular resolution to calculate how far away we can read the book:
Distance = (height of book) / tan(θ)
Substituting the values into the formula, we can find the distance at which we can read the book using the Keck telescope in space.