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Astrophysicists describe the redshift of receding astronomical objects using the redshift factor \(z\), defined implicitly in the following equation: $$ \lambda_{\text {observed }} \equiv(1+z) \lambda_{\text {emitted }} $$ Here \(\lambda_{\text {observed }}\) is the wavelength of light observed from Earth, while \(\lambda_{\text {emitted }}\) is the wavelength of the light emitted from the source as measured in the rest frame of the source. The emitted wavelength is known if one knows the emitting atom, identified from the pattern of different wavelengths characteristic of that atom. Astrophysicists measuring the redshifts of light from extremely remote quasars calculate a \(z\) -factor in the neighborhood of \(z \approx 6 .\) Use the Dopplershift equations of special relativity to determine how fast such quasars are moving away from Earth. Note: Actually, for such distant objects the unmodified Doppler shift formula of special relativity does not apply. Instead, one thinks of the space between Earth and the source expanding as the universe expands; the wavelength of the light expands with this expansion of the universe as it travels from the source quasar to us.

Step by step solution

01

Understand Redshift and its Equation

The redshift factor, denoted by \(z\), can be expressed as \(\lambda_{\text{observed}} = (1+z)\lambda_{\text{emitted}}\). For a quasar with a redshift factor of \(z \approx 6\), this implies that the observed wavelength is 7 times the emitted wavelength.

02

Relate Redshift to Velocity

For determining the velocity of the quasar relative to Earth, use the relativistic Doppler shift formula: \[1 + z = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\]where \(v\) is the velocity of the quasar moving away from Earth, and \(c\) is the speed of light.

03

Substitute Redshift Value

Given \(z = 6\), substitute this value into the Doppler shift equation:\[1 + 6 = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\]\[7 = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\]

04

Square Both Sides

Square both sides to eliminate the square root:\[49 = \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}\]

05

Isolate \(\frac{v}{c}\)

Rearrange to isolate \(\frac{v}{c}\):\[49(1 - \frac{v}{c}) = 1 + \frac{v}{c}\]\[49 - 49\frac{v}{c} = 1 + \frac{v}{c}\]

06

Combine Like Terms

Combine like terms to solve for \(\frac{v}{c}\):\[49 - 1 = 49\frac{v}{c} + \frac{v}{c}\]\[48 = 50\frac{v}{c}\]\[\frac{v}{c} = \frac{48}{50} = 0.96\]

07

Calculate the Velocity

Multiply both sides by \(c\) to find the velocity \(v\):\[v = 0.96c\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

relativistic Doppler shift

The relativistic Doppler shift comes into play when objects moving at speeds close to the speed of light emit or reflect light. When we observe such objects, like quasars, their light appears shifted to different wavelengths due to their high velocities. The shift can be towards the blue end (blueshift) if the object is moving closer to us or towards the red end (redshift) if it is moving away from us.

quasars

Quasars are incredibly luminous objects powered by supermassive black holes at the centers of distant galaxies. They emit immense amounts of energy, often outshining their entire host galaxies. Because of their brightness and distance, studying quasars allows astrophysicists to understand the universe's early stages and the behavior of matter under extreme conditions. The light we observe from quasars has often traveled billions of years to reach us.

speed of light

The speed of light, denoted by the symbol \(c\), is a fundamental constant in physics. It is the speed at which light and all other electromagnetic waves travel in a vacuum, approximately 299,792 kilometers per second (km/s). This speed forms the basis for many equations in relativity, including the Doppler shift formulas used to determine the velocities of distant astronomical objects.

expansion of the universe

The universe has been expanding since the Big Bang. This expansion means that galaxies and other distant objects are moving away from us. The more distant an object is, the faster it seems to move away. This phenomenon is captured in Hubble's Law, which states that the velocity at which a galaxy recedes is proportional to its distance from us. The expanding universe also stretches the light traveling through it, causing redshift as the wavelength of the light increases over time.