I dusted my memory off and as I went along I generated a poor man's, or straw man's possibly, O-C diagram, which is the traditional route.

I think I've cobbled it together in a workable way.

Now, half of this is mindset, so if you stop thinking of Mira stars having a period and instead think of them having an interval between two states you get nearer.

This allows for the inherent scatter (look up the word stochastic) in the 'intervals' not being mistaken for actual periods (especially the visual ones).

This is a fairly simple and reasonably symmetric Mira lightcurve as such things go so doesn't lead to much confusion.

I got the data from a Swedish Astronomy organisation website as there is a person who likes to do this there but doesn't seem to quite understand how an O-C works in my personal view because he tends to stick to the GCVS elements and think there is period change at times when it is actually just the wrong interval length being used. A true period change in the trend displays as an arc, a slanty line means you've just inputted a wrong mean interval. The dates of maximum he quotes may well include stuff from an old AAVSO publication, the trouble with these is you have to take care because some are "extrapolated", that is when a maximum is not observed for a particular Mira they draw in a best fit line from the nearest available bits of lightcurve and use that to guess a maximum, but as you've said M-m isn't always 0.5 for Mira stars. T UMa though is pretty much circumpolar for Northerners so should have had good cover, including WWII where UK and European observations tend to be somewhat rarer for likely obvious reasons, however USA and Canada ones were not so affected. I filled in the last few cycles since the online data using the aforementioned BAA photometric database lightcurve and just guesstimating maximum to a day (Julian days), and why not?

Anyway, I'm not going to bog this down with blurb, I'm going to do it the simplest way. There was a cycle missing in the data at the Swedish website way back early on (cycle is simply a running number between events here so you know they are continuous events and you don't end up with double or triple periods sometimes) so I ditched the first half a dozen maxima and started on the next, the rest being continuous until now (given the four recent maxima times I added). I wanted it simple, gaps cause complications and extra guessing (if there are three maxima missing you have to not get confused on your cycle numbers and end up with that gap interval divided by three to get a 'period' and then you have three identical interval lengths one after the other which is smoothing out what really happens in reality.

I took all the intervals and added them up and found the average, the mean, I also took the median ("middling") and mode (most frequent) values simply because when they are near each other in value you have a rough idea how symmetric and regular the system is if these values are close to each other. I then take the first maximum listed, call it the base epoch, and thus it becomes cycle 0 as that is the one we are going to be measuring from (some people take the last maximum, some work out base epoch more rigorously, I just prefer using the first because you have to anyway to generate an O-C). Then for each maximum the interval between it and the previous one is taken. That gives the average. The dates of Maxima from the list is called the OBSERVED. Then using the base epoch, this mean period and the cycle number you CALCULATE the model dates of Maxima given an ideal lightcurve and fixed period.

That's where you get your O-C values. You plot these against the cycle numbers, as shown on the chart. Then you use linear regression (I think that's what it is called) if you are clever and good at maths, but I just do it the simple way. In an O-C plot you use the spreadsheet feature to add a linear trendline to the plot (see the chart in the attached file). You fix the intercept to pass through 0 O-C. You know the 'period' is near to the average interval calculated from all the intermaximal intervals so you set the CALCULATE column to use values you input by hand very near to that value (marked as the O-C Period in the spreadsheet) and you just keep twiddling the average value until the trend line on the chart goes absolutely flat. You've locked it to intercept 0, to avoid offsets and get a true zero point, you're still using the cycle number and base epoch to calculate, but you're simply tweaking the period by hand.

For example the average of 256.5 days for the intervals between maxima gave a sloping line, I tweaked the decimal values up and down, jumping fairly large at first, the slope changing orientiation when the value was too big or too small and eventually ended up with 256.568 days as the best "period". This isn't a real period, it's the "on average" interval because we are using the idea that the longer the run of data we have the more scatter will smooth out and the better things will be on average when it comes to results.

Well, what next? Cycle number here is unofficial, I just set up a running number to enable doing the sums, although the running number does tally the number of maxima to have occurred since the initial base one (else the sums would give daft answers). It so happens that because I did this the run added up to 226 times of maxima since the base one.

To do an O-C you need to use Julian Days to avoid messy arithmetic. Spreadsheets have an internal running number for dates, the number one being a fixed date. So if you can find out the spreadsheet number of a set date, find the Julian Day number of that same date, then for any cycle number a date can be predicted.

Midnight 25/2/2023 turned out to be JD 2460000.5 and all I then had to do was find the number of days from the predicted Julian day of the next maximum in the software, take 2460000.5 from it, add that numer to the internal spreadsheet running number for 25/2/2023, and set the output cell to date format.

Predictions of maximum date in JD are

base epoch plus (cycle number multiplied by O-C Period)

The only bit you need to use and change in the attached spreadsheet is the cell next to the one labelled ENTER CYCLE No.

Currently it says 227, which is the future maximum date as the last measured one was cycle 226 which was the last time there was a maximum from now.

That calculates the next maximum date as being Friday 28th June 2024.

Notice the spreadsheet also gives the standard deviation of all the interval lengths. That is around 8 days, +/- 8 days (20th June to 6th July 2024) and notice that I stated in an earlier post that Miras tend to have maxima scattered by around +/- 10 days on average.

However, also notice that I give a maximum and minimum interval between maxima. These are real intervals, measured between maxima. They range from 229 days to 299 days, both of which are a long way from the average 256.5 days and well outside the range of 248 and 264 ish days the standard deviation suggests.

This is why other posters have told you to start as far as thirty days earlier than the predicted dates.

The simple reason is that the stars just haven't read the books so don't know what to do to behave as we'd like.

The more valid reason is that these are giant stars, with deep and tenuous atmospheres and not regular clocks, slightest thing can lead to a big difference, and they are 3D objects too however the light comes from the surface of a sphere facing use which is 2D more or less, so small things don't make linear changes, they act as the square or cube (or inverse thereof) making larger changes than in the linear case. Then there's the aspect of the opacque molecular metal oxides that condense out and block the visual light. All these plus the inherent stochastic scatter of the whole thing leads to large outliers at times with intervals kicking off from the average. Traditionally intervals between maxima or minima are used for O-C, and things don't really improve if you use max to min or min to max intervals.

So you have to allow for the occassional completely off interval length.

I'll attach the spreadsheet for you to play with, remember just change the number next to enter cycle no., currently set at 227, increasing by one to get the next prediction (ie 228) and onwards to get predictions after that.

I'll leave it up for a few days but I tend to delete my attachments after a week mostly based on my personal general online data use policy. (I'd probably delete my posts after a month if fora let me).

Remember, you don't have to understand it, the formulae and graph are left in in case you want to stare at them later but all you need to do is change the cycle number. By the way, it won't work far into the past (ie smaller cycle numbers), the JD prediction will be okay but the calendar date friendly bit will not work prioer to 25/02/2023.