Question

Astronomers observe a spectral analysis of a distant star where a particular element has a spectral line with a wavelength of 663 nm. In the laboratory, the same element has a spectral line with a wavelength of 645 nm.
a. How fast is the star moving, and in what direction is it moving?

b. The same element on Earth’s surface is observed from a space shuttle orbiting at 7800 m/s,
350 km above Earth. Is the same shift in the spectral line seen from the space shuttle? Why or why not?

Answer 1

Answer:

(a). The velocity of star is $8.1\times10^{6}\ m/s$ and the direction of star toward the earth.

(b). The shift is 0.0168 nm.

Explanation:

Given that,

Wavelength of spectral line = 663 nm

Wavelength of spectral line in lab = 645 nm

(a). We need to calculate the velocity

Using doppler's effect

$\Delta \lambda=\dfrac{v}{c}\lambda$

Where, $\Delta\lambda$= change in wavelength

v = velocity

c = speed of light

Put the value into the formula

$663-645=\dfrac{v}{3\times10^{8}}\times663$

$v=\dfrac{3\times10^{8}(663-645)}{663}$

$v=8144796.38\ m/s$

$v=8.1\times10^{6}\ m/s$

The direction of star toward the earth.

(b). Speed = 7800 m/s

We need to calculate the shift

Using formula of shift

$\Delta \lambda=\lambda(\sqrt{\dfrac{1+\dfrac{v}{c}}{1-\dfrac{v}{c}}}-1)$

Put the value into the formula

$\Delta \lambda=645\times(\sqrt{\dfrac{1+\dfrac{7800}{3\times10^{8}}}{1-\dfrac{7800}{3\times10^{8}}}}-1)$

$\Delta\lambda=0.0168\ nm$

This shift is small compare to the the movement of Earth around the sun.

Hence, (a). The velocity of star is $8.1\times10^{6}\ m/s$ and the direction of star toward the earth.

(b). The shift is 0.0168 nm.