NASAs Spitzer Space Telescope was launched in 2003 to detect

NASA’s Spitzer Space Telescope was launched in 2003 to detect infrared radiation. The InfraRed Spectrograph (IRS) was able to observe spectral lines from 5.13 to 39.90 micrometers. Astronomers want to detect hydrogen lines in the Humphries series (n=6) from a red giant star. (a) What lines can be observed by Spitzer’s IRS?

(b) Suppose the detector measures four lines with wavelengths of 12.496, 7.578, 5.967 and 5.181 micrometers. Why are these wavelengths different from those found in part (a)?
(c) How fast is the star moving with respect to us?

Solution

a) What lines can be observed by Spitzer’s IRS?

Hydrogen spectral series are given by

1/ = R(1/n_f² -1/n_i² ) where R = 1.0973732e+7 /m

n_f =6   , n_i = 7,8,9,10,11 …………

n_f 6, n_i = 7

1/ = (1/36-1/49)*1.0973732e+7

   = 12.365   m

the series are 12.365, 7.498, 5.905, 5.125, 4.669    m

due to the limitation of IRS only 12.365, 7.489, 5.905, 5.125 lines can be observed.

b) Suppose the detector measures four lines with wavelengths of 12.496, 7.578, 5.967 and 5.181 micrometers. Why are these wavelengths different from those found in part (a)?

As the star is moving with a relativistic velocity there will be a frequency shift on the observed frequency as given by

   f_obs = sqrt{(1+v/c)/(1-v/c)}*f_src

where the light source is approaching the observer, in our case the star is receding from the earth and hence v is –ve. We can right the above expression as

f_obs = sqrt{(1-v/c)/(1+v/c)}*f_src
the observed frequency will be less than the actual frequency emitted by the source and the wave length will be higher

c) How fast is the star moving with respect to us?

c/12.496 = sqrt{(1-v/c)/(1+v/c)}*c/12.365

v/c = (1-0.979)/1.979 = 0.01c

the start is receding from earth at a speed of 0.01c