Equations needed:
(1.) Wien's Law: T= 3 x 106nm/l
The temperature is in Kelvin (Degrees
Celsius +273.15).
For the rest of the problem set you will need to consider intensity:
Intensity is the power per area.
I=P/A
I, the intensity, is the power (Watts)
per unit area (m2).
Note: Power and Luminosity are the same thing. (We tend to call
the power of a luminous object luminosity)
Intensity from a "black body":
(2.) I=constant T4
note that T4 is TxTxTxT !!!!
(the constant, named sigma, or s in the book pg. 366, is equal to 5.67 x 10-8 Watt / (m2 K4).)
We don't really need to know what the constant is. Rather, for two hot objects, indexed 1 and 2, of different areas with the same temp we get:
(2.1) P1/P2 = Area1 / Area2
And both constant and temp. cancel out.
For the two balls with the same areas, but different temperatures:
(2.2) P1/P2 = T14/T24
And both constant and area cancel out.
If the two objects have DIFFERENT temperature AND DIFFERENT area the ratio of intensities is
(2.3) P1/P2 = Area1T14/(Area2T24).
The area of a sphere is:
(3) A=4pr2
Where r is the radius of the sphere.
Problem #2: I have two spheres, one
with a radius of 0.2 m and the other with a radius of 0.5 m. I heat the
small one to a temperature of 2400 K, and the big one to a temperature
of of 2100 K.
a) What is the ratio of the intensities of the two spheres?
Answer: 0.586
b) What is the ratio of the total power emitted by these
spheres? Answer: 3.66
Problem #3: Calculate the total power emitted
at all wavelengths by a star whose surface temperature is 7300 K, and whose
radius is 2.5 solar radii. (You may assume that the star radiates as a
blackbody.) Answer in units of solar power and solar radius.
Problem #4: Calculate the radius of a Main
Sequence B0 star (which has a surface temperature of 3 x 104
K, and a luminosity of 1 x 103 Lo, where Lo means "solar luminosities").
Answer in units of solar radius.
Problem #5: (For extra credit only, so that you may
get 12.5/10 for this HW assignment) A spherical planet of radius
6 x 106 m and surface area 4.5 x 1014 m^2 is 1.5
x 1011 m from a G2 star, with 1 Lo. One side of the star is
always facing the sun. Assume that the albedo is 100%, that is: all light
from that star is absorbed by the planet, and then re-emitted as black-body
radiation.
a. What is the intensity (power/area) of light from the star at the
surface of the planet?
b. What is the surface temperature of that planet?
c. What is the total luminosity of the reflected light in units of
Lo?
d. What may be the implications of this result for the search for extra-solar
planets similar to earth?