The spectrum of light from the galaxy M100 has a prominent dark line at a wavelength of 592.2 nm. The same spectral line observed in a lab on earth has a wavelength of 589.4 nm. How fast is M100 moving relative to earth? (the speed of light = 3.00 x 108ms-1)
(delta L)/L = v/c
L is the wavelength. v is the galaxy radial velocity (in the direction away from earth). Velocity components normal to that direction cannot be measured spectroscopically.
Since the wavelength shift is to the red (longer wavelengths), it is moving away.
Use the equaition provided to calculate V
To determine the speed at which the galaxy M100 is moving relative to the Earth, we can use the concept of the Doppler effect.
The Doppler effect describes how the wavelength of light changes when an object emitting that light is in motion relative to an observer. When an object is moving towards an observer, the observed wavelength appears shorter, i.e., it is shifted towards the blue end of the spectrum (known as a blue shift). On the other hand, when an object is moving away from an observer, the observed wavelength appears longer, i.e., it is shifted towards the red end of the spectrum (known as a red shift).
In this case, we are given that the dark line from the galaxy M100 has a wavelength of 592.2 nm when observed from Earth, and the same line observed in a lab on Earth has a wavelength of 589.4 nm. The observed line is shifted towards the red end of the spectrum, implying that the galaxy is moving away from the Earth.
The difference in wavelengths observed is given by:
Δλ = λ_observed - λ_rest
where Δλ is the difference in wavelengths, λ_observed is the observed wavelength, and λ_rest is the rest wavelength.
Therefore, we have:
Δλ = 592.2 nm - 589.4 nm = 2.8 nm
Now, to calculate the speed at which M100 is moving relative to Earth, we can use the formula for the Doppler effect:
Δλ/λ_rest = v/c
where Δλ is the observed shift in wavelength, λ_rest is the rest wavelength, v is the velocity of the object (M100 in this case), and c is the speed of light.
Rearranging the equation to solve for v, we get:
v = (Δλ/λ_rest) * c
Plugging in the given values, we get:
v = (2.8 nm / 589.4 nm) * 3.00 x 10^8 m/s
Simplifying the equation, we find:
v ≈ 1.430 x 10^6 m/s
Therefore, the galaxy M100 is moving away from Earth at a speed of approximately 1.430 x 10^6 m/s.