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Source: http://katjapoppenhaeger.com/?p=457

I managed to trace this graph back to http://katjapoppenhaeger.com/?p=457. It's meant to show how stellar activity declines with time for different star types, and whilst I get the general idea, I don't really understand what either axis means. Y is presumably some measurement of activity, and X is probably age, but the logarithmic part is really unclear. Is the X axis using E+ (1e+1 = 10) numbers?

I'm really into realistic worldbuilding, but I was never really good at maths...

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    $\begingroup$ It's true that putting the unit inside the log parenthesis, moreover after a slash, is a bit misleading... "Log(stellar age) (yr)" would have been much clearer. $\endgroup$ Apr 19, 2023 at 11:08
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    $\begingroup$ Except logarithms need to be dimensionless. So putting the unit in the bracket after a / is correct as your dividing by a standard unit, in this case 1 year. $\endgroup$
    – Rob
    Apr 19, 2023 at 14:02

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The axis description is quite clear, especially if one takes the image description from your link:

On the x-axis you have the 10-based logarithm of the stellar age measured in years. Thus 0 means 1 year and 6 means 1 million years, 8 means 100 million and 9 means 1 billion years (basically the number on the axis is the number of zeros in the age).

The y-axis is a measure of the some luminosity per unit surface of the star. Your source specifies it more clearly than the axis description shown here: it shows the logarithm of the x-ray luminosity scaled for stellar surface which is commonly a proxy for the (magnetic) activity of the star. The there referenced paper tells us more precisely that the unit is the logarithm of the energy output divided by the cross-section of our Sun in erg/s/$R_{Sun}^2$. 33 would mean it emits as much energy in the X-ray as our Sun emits over all wavelengths, 32 means 1/10 of that and 31 means 1/100 etc.

In essence this graph tries to tell: the star type is not of so much importance in the star's activity (though you see some slight dependence in the colour coding). The star's age is more important. Especially stars older than 1 billion years will show decreased activity or luminosity in the X-ray. The point of her paper seems to be that she found a revised calibration to relate activity to age - and that activity with their measurements now can hence be used more accurately than before to infer stellar age as activity is possible to measure by observation.

You can estimate any number between the points by simply applying the reverse of the logarithm: if e.g. $\log_{10}(x) = 9.5$, then $x = 10^{9.5} = 3.16$ billion.

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    $\begingroup$ Maybe mention that the units of the y-axis are also base-10 logarithms (of X-ray luminosity)? $\endgroup$ Apr 19, 2023 at 7:55
  • $\begingroup$ Thank-you! Would I be able to determine points between the numbers? $\endgroup$
    – Kazon
    Apr 19, 2023 at 14:25
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    $\begingroup$ The y-axis is log to base 10 of X-ray luminosity per solar surface area. (in ergs/s). $\endgroup$
    – ProfRob
    Apr 19, 2023 at 15:45
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    $\begingroup$ Thanks @ PeterErwin @ ProfRob, I applied your suggestions for amendment. $\endgroup$ Apr 19, 2023 at 16:50
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One purpose of plotting a graph of $\log(y)$ against $\log(x)$ is to look for a relationship in the form $y = Ax^b$.

Look at the black line drawn on the graph, it is straight and we can measure it's gradient to work out a linear formula in terms of log(Lx) and log(stellar age). I can eyeball this formula (with a little help from geogebra) as $$\log(L_x) = -2.8 \log(\text{Age})+ 54.7$$

Now these are base 10 logarithms, so applying 10-to-the power on both sides gives $$10^{\log(L_x)} = 10^{-2.8 \log(\text{Age})+ 54.7}$$

and simplifying, since $10^{a+b}=10^a\times 10^b$ and $10^{log x}=x$ gives:

$$L_x =\text{Age}^{-2.8}\times 10^{54.7} = \frac{10^{54.7}}{\text{Age}^{2.8}}$$

This is their proposed model for the x-ray luminosity normalised for stellar surface, for stars between 1 billion and 10 billion years old.

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