130_notes.dvi

2.6 Examples

2.6.1 The Solar Temperature*

Estimate the solar temperature using

• the solar radiation intensity on the earth of 1.4 kilowatts per squaremeter. (rsun= 7× 108
  m,dsun= 1. 5 × 1011 m)


• and the solar spectrum which peaks at about 500 nm.

First we compute the power radiated per unit area on the solar surface.

R= (1400W/m^2 )(4πd^2 sun)/(4πrsun^2 ) = 6. 4 × 107 W/m^2

We compare this to the expectation as a function of temperature.

R(T) = (5. 67 × 10 −^8 W/m^2 /◦K^4 ) T^4

and get

T^4 =

6. 4 × 107

5. 67 × 10 −^8

T = 5800◦K

Lets assume thatE(λ,T) peaks at 500 nm as one of the graphs shows. We need to transform
E(ν,T). Rememberf(ν)dν=g(λ)dλfor distribution functions.

E(ν,T) =

2 πν^2

c^2

hν

ehν/kT− 1

E(λ,T) =

∣

∣

∣

∣

dν

dλ

∣

∣

∣

∣

2 πν^2

c^2

hν

ehν/kT− 1

=

ν^2

c

2 πν^2

c^2

hν

ehν/kT− 1

=

2 πν^4

c^3

hν

ehν/kT− 1

This peaks when
ν^5
ehν/kT− 1
is maximum.

5 ν^4

ehν/kT− 1

−

ν^5 (h/kT)ehν/kT

(ehν/kT−1)^2

= 0

5

ehν/kT− 1

=

ν(h/kT)ehν/kT

(ehν/kT−1)^2

5(ehν/kT−1)

ehν/kT

=hν/kT

5(1−e−hν/kT) =hν/kT