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Doppler Broadening``

Problem

Calculate the Doppler broadening of the iron lines at the surface of the Sun.

 

Hints:

Calculate the most probable speed of the iron atom at the Sun's surface temperature of 5780 K, according to a Maxwellian distribution.

Divide this result by the speed of light to calculate the Doppler broadening for iron on the solar surface.

 

Data

Mass of the Hydrogen Atom

m[H] := 1.67*10^(-27)*Unit('kg')

 

Mass of Iron Atom

m[Fe] := 56*m[H]

 

Boltzmann Constant 

k := 1.380650277*10^(-23)*Unit('J')/Unit('K')

 

Temperature of the Solar Surface

T[Sun] := 5780*Unit('K')

 

Speed of Light

c := 3*10^8*Unit('m')/Unit('s')NULL

Useful Equations

Particle Velocity, Maxwellian Distribution

v[p] = (2*k*T/m)^(1/2)

 

Formula for Doppler Shift or Broadening

`Δλ`/lambda = v/c

Solution

The mass of the iron atom is 56 times that of the hydrogen atom (56 * 1.67*10^(-27) kg). Calculate the most probable speed of the iron atom at the Sun's surface temperature of 5780 K, according to a Maxwellian distribution:

NULL

evalf(v[p] = (2*k*T[Sun]/m[Fe])^(1/2))

v[p] = 1306.376953*(Units:-Unit('J')/Units:-Unit('kg'))^(1/2)

(4.1)

"(->)"

v[p] = 1306.376953*Units:-Unit(('m')/('s'))

(4.2)

NULL

``

Divide this result by the speed of light to calculate the Doppler broadening for iron on the solar surface:

``

`Δλ` = simplify(evalf(rhs(v[p] = 1306.376953*Units:-Unit('m'/'s'))/c))

`Δλ` = 0.4354589843e-5

(4.3)

NULL

``