# Hubble constant from Time delays

1. Dec 6, 2013

### egsid

Hello, everyone it seems to me that people understand how to do this problem but I am struggling to solve it.
1. The problem statement, all variables and given/known data
A quasar with redshift 0.3 is gravitationally lensed into two images by an elliptical galaxy at redshift 0.18. There are two images of the quasar which are separated from the center of the galaxy by 1.10 arcsecs and 1.60 arcsecs on opposite sides. One of the quasar images flares up in intensity by 1 mag. and 16.7 days later the second quasar image flares by the same amount and with the same time profile. Assume the quasar is directly behind the galaxy and that the light paths are straight lines from the quasar to the point nearest the galaxy, then bend, and then are again straight from the point nearest the galaxy to the Earth. From this information, calculate Hubble Constant.

2. Relevant equations

I think this is what is my problem.
I think maybe v = H0*d is the relevant equation. The angles given may help determine what d is which in some combination will help determine Hubble constant?

3. The attempt at a solution

So I drew my diagram and labeled it with the relevant details. I thought you had to calculate the velocity of the quasar and galaxy so I used the equation 1+z = sqrt((c+v)/(c-v)) to find them. Obviously the time delay is 16.7 days but I don't know how to use this to calculate the Hubble Constant. Also, I don't know if the fact that the intensity "flares up in intensity by 1 mag." is relevant.

Thanks

Last edited: Dec 6, 2013
2. Dec 11, 2013

### haruspex

I've no background in relativity theory, so I may be off the mark here, but it looks like a simple geometry problem.
You have a straight line E (Earth), G (Galaxy), Q (Quasar). At right angles to this are the lines AG, GB to the points where the light paths are taken to bend. You know the ratio EG:GQ, and the angles GEA, GEB. That gives you the ratios of all the lengths in the diagram. The delay tells you the difference in length between the paths QAE, QBE, so that fixes all lengths absolutely.