Estimate the observed wavelength for the $656 \mathrm {~nm}$ line in the spectrum of a star which is 100 million light-years from us if $H = 20 \frac { \mathrm { km } / \mathrm { s } } { \mathrm { Mly } }$ ? $\left( c = 3.00 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$
A) $660 \mathrm {~nm}$
B) $640 \mathrm {~nm}$
C) $690 \mathrm {~nm}$
D) $620 \mathrm {~nm}$

Question

Quizplus · Accepted Answer

The observed wavelength can be calculated using the formula for redshift, $z = \frac{\Delta \lambda}{\lambda} = \frac{v}{c}$, where $v$ is the velocity of the star moving away from us, $c$ is the speed of light, and $\lambda$ is the original wavelength. Given $H = 20 \frac { \mathrm { km } / \mathrm { s } } { \mathrm { Mly } }$ and the distance is 100 Mly, the velocity $v = H \times \text{distance} = 20 \times 100 = 2000 \, \mathrm{km/s}$. Converting $v$ to m/s gives $2000 \times 10^3 \, \mathrm{m/s}$. Thus, $z = \frac{2000 \times 10^3}{3.00 \times 10^8} = \frac{2}{300}$. The observed wavelength $\lambda' = \lambda(1 + z) = 656(1 + \frac{2}{300}) = 656 \times \frac{302}{300} \approx 660 \, \mathrm{nm}$.

Estimate the Observed Wavelength for The 656 nm656 \mathrm {~nm}656 nm Line in the Spectrum of a Star Which Is 100

Estimate the Observed Wavelength for The $656 \mathrm {~nm}$ Line in the Spectrum of a Star Which Is 100