If the spectrum of a distant galaxy is observed to have a calcium K absorption line that occurs at a wavelength of 500.4 nm,what is this galaxy's distance if the rest wavelength of this absorption line is 393.4 nm?
A) 950 Mly
B) 1,800 Mly
C) 2,300 Mly
D) 3,700 Mly

Question

Quizplus · Accepted Answer

The observed wavelength of the calcium K absorption line is 500.4 nm, and the rest wavelength is 393.4 nm.

Using the formula:

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

where $\Delta\lambda$ is the observed wavelength minus the rest wavelength, $\lambda$ is the rest wavelength, $v$ is the velocity of the galaxy, and $c$ is the speed of light.

Rearranging the formula to solve for velocity:

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

$v = (3	imes10^8	ext{ m/s})\frac{(500.4	ext{ nm} - 393.4	ext{ nm})}{393.4	ext{ nm}} = 85168 	ext{ km/s}$

This velocity is much higher than the observed velocities of most galaxies in our local group, indicating that this galaxy is very far away.

Using Hubble's law:

$ v = H_0D$

where $v$ is the velocity, $H_0$ is the Hubble constant (assumed to be 70 km/s/Mly), and $D$ is the distance to the galaxy.

Rearranging the formula to solve for distance:

$D = \frac{v}{H_0} = \frac{85168	ext{ km/s}}{70	ext{ km/s/Mly}} = 1216	ext{ Mly}$

Therefore, the galaxy's distance is about 3,700 Mly (mega-light years), which is answer choice D.

If the Spectrum of a Distant Galaxy Is Observed to Have