# Glow in the dark fading

I am doing some experiments with "glow in the dark stars". from basic theory I expected the light to drop off at:-

$$I = I_0 e^{-t/T}$$

However, on measuring this with a Lux sensor I find the drop off is linear. Is there something I am not understanding about this?

• What is the ratio of your measured maximum and minimum illuminances? How many points in time did you measure? – Ruslan Aug 21 at 14:39

This is not a complete answer but rather some things you could look at to help to debug the problem.

First of all are you sure the meter you are using has a scale which reads linearly rather than logarithmically? A logarithmic scale would be far more useful for anything which wants to have a large range of readings, or which is meant to be used to judge subjective brightness, as human vision is approximately logarithmic, or photography where light is assessed logarithmically, and would nicely explain your problem.

Secondly are you reading over a long enough period of time? Remember that

$$e^{t/\tau} = 1 - \frac{t}{\tau} + \frac{t^2}{2\tau^2} - \ldots$$

So, over short times, this looks linear because the higher-order terms have not had a large enough effect yet. If that's the case then you simply need to measure for longer. I suspect this might be the problem.

Finally if you are convinced that it really does look linear here's a thing you can do which will test whether the hypothesis is correct (which, as you've said, it's almost certainly not). If it's linear then the intensity looks like $$I = I_0(1 - t/\tau)$$, and you can determine $$I_0$$ and more importantly $$\tau$$. Well, when $$t = \tau$$ then $$I = 0$$. So, now, set up some of the things in the dark and wait until $$t = \tau$$ & measure the intensity. If it's not zero then the decay is not linear.

• Perhaps worth noting that a logarithmic scale also is useful if your instrument is supposed to show how bright the source appears to human eyes. Eyes have a logarithmic response to light intensity. – Solomon Slow Aug 21 at 13:31
• @SolomonSlow: good point, I've amended it – tfb Aug 21 at 14:20
• Thanks, yes indeed I was correct. My error was in not getting the Excel chart correct and plotting the same series against itself which will of course give you a straight line. Thanks for your input, it pushed me over the line. – Grumpy-Mike Aug 27 at 19:28