# Measuring SLM using a quasar jet image at different epochs in time

Though I include step-by-step instructions here, the results are not shown. Try these commands by yourself and find out your own result.

## Contents

## Find the distance to 3C 279

Set up the constant parameters first:

The Hubble constant, in units of **km/s/Mpc**

H0 = ... ;

the redshift of 3C 279 (redshift has no unit!)

z = ... ;

the speed of light, in units of **m/s**

c = ... ;

The receding velocity can be calculated from the redshift as and . Doing some calculations you'll solve ** v_rec**:

v_rec = ...

v_rec = ...

This is in units of **m/s**. To calculate the distance using the Hubble constant, we need to divide ** v** by 1000 to get v in units of

**km/s**. Then we can get D in

**Mpc**:

D = ...

D = ...

## The scale between angle on the sky and the distance

*If you are not familiar with angular diameter, check the wiki page: Angular Diameter*

The real "ruler" on the sky at distance ** D** is the angle we observed times the distance

**, while the angle is measured in**

`D`**rad**. The conversion factor from

**deg**to

**rad**is

deg2rad = ... ;

Therefore, 1 **arcsec** = 1 **deg** / (60*60) is (in units of **rad**)

asc = deg2rad / ... ;

Therefore the conversion factor between 1 **arcsec** on sky and distance at 3C 279 (in **pc**) is (recall that our ** D** is in units of

**Mpc**= 1e6

**pc**)

as2pc = asc * ...

as2pc = ...

## Apparent velocity of the knot

Using the image, I estimate the brightest knot moved from about **A milliarcsec** away from the central source to about **B milliarcsec** in **N years**. The velocity in the sky is thus

vSky = (B - A)/N;

This is in units of **milliarcsec / years**, which is not very useful. We want to convert it to our familiar unit system **m/s**.

First, convert the time unit from **year** to **second**:

yr2s = ... ;

so the velocity in **milliarcsec/second** is

vma = ...

vma = ...

and in **arcsec/second**:

va = vma * ...

va = ...

Then let's convert **milliarcsec** to the real ruler on the sky, **pc**:

vpc = va * ...

vpc = ...

And the conversion factor from **pc** to **meters**: 1*pc* = 3.086e16 **m**

pc2m = 3.086e16;

Then we have the measured velocity in **m / s**:

v = vpc * ...

v = ...

This velocity is much greater than the speed of light!

## The true velocity

But, we know this is not true velocity. If we assume an inclination angle of the jet along which the knot moves to the line of sight which maximizes apparent velocity, then we get a lower limit on the actual velocity of the knot in the rest frame of the Quasar:

beta = ...

beta = ...

which means **v_real** is

v_real = beta*c

v_real = ...

very close, but smaller to the speed of light!