A He-Ne laser onboard a spaceship moving toward a remote...

A He-Ne laser onboard a spaceship moving toward a remote space station emits a beam of red light directed toward the space station. The wavelength of the light in the beam, as measured by a wavelength meter on the spaceship, is $632.8 \mathrm{nm}$. If the astronauts on the space station see the beam as a blue beam of light with a measured wavelength of $514.5 \mathrm{nm}$, what is the relative speed of the spaceship with respect to the space station? What is the red-shift parameter, $z$, in this case?

Key Concepts

Relativistic Doppler Effect

This concept describes how the wavelength (and therefore frequency) of light changes when there is relative motion between the source and the observer, particularly at speeds that are a significant fraction of the speed of light. In the context of special relativity, the formula for the Doppler effect factors in time dilation, leading to an observed shift that can result in either a redshift or a blueshift depending on whether the source and observer are moving apart or towards each other.

Redshift Parameter (z)

The redshift parameter is a dimensionless quantity defined as the fractional change in the wavelength of light, calculated as z = (?_observed - ?_emitted)/?_emitted. Although often associated with objects moving away (resulting in a positive z, or redshift), the same formulation applies when the object is moving towards the observer, in which case z would be negative (indicating a blueshift). It provides a measure of how much the motion of the source or the expansion of space has altered the observed wavelengths.

Relative Velocity in Special Relativity

In scenarios involving high velocities close to the speed of light, calculating the relative velocity between source and observer requires the use of special relativistic formulas rather than classical approximations. These formulas relate the observed wavelength shifts to the speed of the moving source, often denoted as a fraction of the speed of light (? = v/c). They account for effects such as time dilation and length contraction, ensuring that the computed velocity remains consistent with the postulates of special relativity.

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