Slope of log plot of luminosity-velocity in Tulley Fisher relation

My partner and I did some computation on the Tully Fisher relation at uni.

In that, the slope of the logarithm plot between the luminosity and rotational velocity, was 2.6.

Now, that slope corresponds to the exponent to which $$v_{rotation}$$ is raised to. i.e.

$$L=Av_{rotation}^b$$

where $$b$$ is the exponent.

So, $$b$$ officially is close to 4. And I have used up a sample size of over 10 galaxies. And the plot looked pretty shabby honestly. Is our result poor?

• No. Is the data authentic enough to actually be legitimiate? Because only four out of those points actually meet the red line. And so, I was wondering if this was in fact a bad fit, then the uncertainties of the data points have to be re done. – Karthik Mar 5 '20 at 22:10
• Okay thanks! I think that an answer will end up explaining how to calculate the uncertainty in the slope and/or a "goodness of fit" metric and recommend how you can decide for yourself if those indicate a result that is good or poor. Science is science, if this is the data, then this is the data. If it fits a line, it fits, if it doesn't it doesn't. We don't "re do" data points to make the fit better, that's a science no-no! :-) – uhoh Mar 5 '20 at 22:24
• @uhoh thanks so much mate. I needed a bit of a breather. Is it normal to produce "not-so-correct" results the first time in astronomy? – Karthik Mar 6 '20 at 3:39
• Data are data. All you need to do is discuss the errors, the goodness of fit and possibly what you can do during further measurements to get data and fit /theory match better. Look what contributed to the errors of the individual data points. Look at the error of the slow of your fit – planetmaker Mar 6 '20 at 10:07
• You don't say what bandpass the luminosity was measured in. Slopes of closer to 3 are seen when using optical (especially blue) luminosities. And given that you have only nine data points, you shouldn't expect a very accurate result anyway. – Peter Erwin Mar 6 '20 at 15:59