Download The Inverse Square Law and Surface Area

The Inverse Square Law and
Surface Area
Calculating Distances to Stars
Measuring Distances
There are several techniques used to
measure distances to stars. The distance
to the very closest stars can be measured
by trigonometric parallax
The diagram shows Earth Orbit around the Sun.
The position of a nearby star changes by a tiny amount over a six month
period. This allows us to use trigonometry to find its distance.
The angles are extremely small.
This direct method is the most accurate way of determining distance
Using The Inverse Square Law.
Every instant a star radiates its energy into space
The energy which was at the surface is distributed at the surface of an
expanding sphere
The amount of electromagnetic energy at every instant emitted by the star
is at the surface of an expanding sphere whose radius is the distance to
the star
D
star
Earth
This energy distribution obeys an inverse square law.
This is because the surface
area of this sphere is 4πD2
The Inverse Square Law
The power received from a star per metre squared at the
Earth is called the intensity (I) of the star’s radiation
This is related to the power of the star in this way
P
I
2
4D
Where D is the radius of the sphere
i.e. the distance from Earth to the star
The Intensity of the Sun
• At Earth the measured power output of the
Sun is 1.3 kWm-2.
• The mean distance to the Sun is 150
million km.
• Calculate the power output of the Sun.
Stars of Known Power Output
• There are several classes of stars with known power
output.
• Stars which have the same surface temperature ( and
spectral characteristics) as the sun all have the same
power output
• We can readily calculate the power output of nearby
stars and classify their power output and compare them
with more distant stars
The following very bright objects of known luminosity can
be identified in distant galaxies
• Cepheid Variable Stars
• Supernovae
The Sun has a power output of 3.91 x 1026W.
Knowing this and knowing its surface temperature allows us to
calculate its surface area using P=σ.AT4
P
A
T 4
This now allows us to calculate the radius of the Sun as A =4πr2.
Calculate the solar radius.
Comparing the Power Output
of the Sun with other Stars
(Using Stefan’s Law: P=σAT4)
Sources Which Have The Same
Power Output As The Sun
Example: calculate the radius of a star which has a surface
temperature of 3 700 degrees and the same power
output as the Sun. What is the diameter of this star
taking the solar diameter as 1 unit?
(Ps = 3.91 x 1026W) (Ds = 1.4 million km)
Apply Stefan’s Law P=σAT4
.
• What is the diameter of a star, the
maximum intensity of which occurs at
625nm
Steps:
1. Use Wien’s law λmax T = 2.9 x 10-3 mK to
calculate the temperature of the star
2. Apply Stefan’s Law to calculate its radius.