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# Spectroscopy

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Measuring the redshift of 3C 273 – The Brightest Quasar

### How to measure the redshift and distance:

To measure the distance and redshift of the quasar we can use the following method: The quasar spectrum image is loaded into a spectral analysis program such as RSpec. Here the quasar spectrum is firstly calibrated against another spectrum of a nearby reference star (with negligible redshift). In my spectrum the Balmer lines for H-Alpha and H-Beta appear prominently as strong emission features and there is also a hint of the H-Gamma line. The observed peak wavelengths (λ) of the emission lines is then identified, and their offset relative to the true emitted wavelengths is calculated. In my spectrum image I measured the following wavelength shifts:H-Alpha: 7674Å

_{(observed)}- 6563Å

_{(emitted)}= 1111Å.

H-Beta: 5692Å

_{(observed)}- 4861Å

_{(emitted)}= 831Å.

H-Gamma: 5083Å

_{(observed)}- 4340Å

_{(emitted)}= 743Å.

To calculate the redshift z we can first use the classical formula:

z = λ

_{(observed)}- λ

_{(emitted)}/ λ

_{(emitted)}

For the three emission line offsets we get an average z = 0.17. This number is significantly higher than the officially reported value of z = 0.158, but this is only because 3C 273 is moving away from us at such great velocity and the classical formula does not take relativistic speeds into account.

We can calculate the relativistic velocity of 3C 273 using the formula v = c * ((z+1)

^{2}-1) / ((z+1)

^{2}+1), and get 46,787 km/s.

This means that 3C 273 is receding from us at nearly 47,000 km/s, which is over 15% of the speed of light! Now we can use the relativistic velocity to calculate a proper relativistic redshift using z = v/c and we get

**z = 0.156**which is very close to the officially reported value of z = 0.158339 in the NASA/IPAC Extragalactic database.

Lastly, using Hubble's law, we can calculate the distance to 3C 273 as d = v / H

_{0}. Assuming the current best estimate for Hubble's constant H

_{0}= 69.8(km/s)/Mpc we get 670.31 Mpc or roughly 2.186 billion light years!

Because of their incredible luminosity, quasars are some of the most distant objects visible to us on Earth. When we observe such distant objects we also see back in time, because their light has taken very long to reach us. So we are really viewing 3C 273 as it appeared over 2 billion years ago.

3C 273 is one of the most luminous quasars known, being 4 trillion times more luminous than the Sun and having an absolute magnitude of −26.7, meaning that if it were only 30 light years from us it would appear as bright in the sky as the Sun!

It is truly a remarkable object and it is fascinating that the distance to a remote quasar can be estimated quite accurately using a modest amateur telescope and a simple diffraction grating!

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