Question

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?

Short answer

Answer: The relative speed of the spaceship concerning the space station is approximately \(9.18 \times 10^7\) m/s, and the redshift parameter is approximately -0.1873.

Step-by-step solution

Writing down the given information

We have the following information: - \(\lambda_{source} = 632.8\) nm - \(\lambda_{obs} = 514.5\) nm

Calculate \(\beta\)

We have the equation: $$\frac{\lambda_{obs}}{\lambda_{source}} = \sqrt{\frac{1 + \beta}{1 - \beta}}$$ We can solve for \(\beta\) by first squaring both sides of the equation: $$\left(\frac{\lambda_{obs}}{\lambda_{source}}\right)^2 = \frac{1 + \beta}{1 - \beta}$$ Now, rearrange the equation so that \(\beta\) is on one side of the equation: $$\beta = \frac{(\frac{\lambda_{obs}}{\lambda_{source}})^2 - 1}{(\frac{\lambda_{obs}}{\lambda_{source}})^2 + 1}$$ Now, plug in the given wavelengths: $$\beta = \frac{(\frac{514.5}{632.8})^2 - 1}{(\frac{514.5}{632.8})^2 + 1}$$ Calculate \(\beta\): $$\beta \approx 0.3059$$

Calculate relative speed

Now that we have the value of \(\beta\), we can find the relative speed of the spaceship concerning the space station by multiplying \(\beta\) by the speed of light: $$v = \beta \times c$$ Since the speed of light, \(c = 3 \times 10^8\) m/s, we have: $$v = 0.3059 \times 3 \times 10^8$$ $$v \approx 9.18 \times 10^7$$ m/s So the relative speed of the spaceship concerning the space station is approximately \(9.18 \times 10^7\) m/s.

Calculate the redshift parameter, \(z\)

To find the redshift parameter, use the equation: $$z = \frac{\lambda_{obs} - \lambda_{source}}{\lambda_{source}}$$ Plugging in the given wavelengths: $$z = \frac{514.5 - 632.8}{632.8}$$ $$z \approx -0.1873$$ Since we have a blueshift instead of a redshift, the value of \(z\) is negative. Therefore, the redshift parameter in this case is approximately -0.1873.

Key concepts

Redshift and Blueshift

The concepts of redshift and blueshift are central to understanding how we perceive the motion of astronomical objects in the universe through the light they emit. Redshift occurs when an object is moving away from the observer, causing the light's wavelength to appear longer and shift towards the red end of the spectrum. In contrast, blueshift happens when an object is moving towards the observer, resulting in a shorter-wavelength light that shifts toward the blue end.

For instance, when a star or galaxy moves away from us, the light undergoes a redshift, which astronomers can use to determine the speed at which that object is receding from the Earth. The greater the redshift, the faster the object is moving away. Similarly, a blueshift signals that the object is approaching. The changes in the light's wavelength are a direct result of the Doppler effect in light, comparable to the changes in sound frequency we experience when a siren passes us on the street.

Relativistic Doppler Effect

The relativistic Doppler effect is an extension of the classical Doppler effect that takes into account the finite speed of light and the effects of special relativity. This becomes significant at velocities that are a substantial fraction of the speed of light. Unlike sound waves, which require a medium to travel through, light waves can travel through the vacuum of space, and their frequency shift obeys different rules when the relative speeds approach the speed of light.

The formula \[\frac{\lambda_{obs}}{\lambda_{source}} = \sqrt{\frac{1 + \beta}{1 - \beta}}\] represents the relativistic Doppler effect, where \(\lambda_{obs}\) and \(\lambda_{source}\) are the observed and emitted wavelengths, respectively. \(\beta\) is the ratio of the object's relative speed to the speed of light. The equation accounts for time dilation and length contraction, which are key aspects of special relativity and affect how we measure the wavelength of light from moving objects.

Speed of Light

The speed of light in a vacuum, denoted by \(c\), is one of the fundamental constants of nature and plays a crucial role in astronomy and physics. It is exactly \(299,792,458\) meters per second. This constant speed means that light from distant stars and galaxies takes significant amounts of time to travel to us, allowing astronomers to look back in time as they observe faraway celestial objects.

It's important to note that, according to Einstein's theory of relativity, nothing can travel faster than the speed of light in a vacuum. This universal speed limit has profound implications for understanding the universe and affects not just the propagation of light, but also how we interpret measurements of distance and time across expansive cosmic distances.

Wavelength Measurement

Measuring the wavelength of light plays a crucial role in understanding astronomical phenomena and the Doppler effect. Wavelength is the distance between successive peaks of a wave, typically measured in meters or its subunits, such as nanometers (nm) for light waves. Astronomers use various instruments, such as spectroscopes, to determine the wavelengths of light received from space. By comparing these observed wavelengths to known values, shifts can be identified and quantified.

To accurately measure the wavelength, factors like the instrument's resolution and the light's intensity are taken into account. Precise measurements enable scientists to calculate important parameters, such as an object's speed and direction of movement relative to the observer, providing deeper insights into the dynamics of the cosmos.