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(I bolded the actual question, below, so if you want to skip the preamble scroll down to the bold lettering)

I am working on a world that is very similar to earth in some ways, exactly the same mass and atmosphere, slightly less water mass. The semimajor axis is about 1.05 AU, Eccentricity is 0.01 or so, with a 31 degree axial tilt. Perihelion is about 5 days after the spring solstice. Not that it's relevant, but there is a single inferior planet and two superior planets, one of which is a gas giant, named hot chi, which does pull the primary around a bit and consequently the sidereal year starts out varying very slightly from one year to the next, because I didn't quite balance the momentum of the system exactly, but it's pretty close and stabilizes after a few decades. The world itself is Home:

Orbit Plot

I would like to determine in approximate terms what the growing season is, presuming earthlike crops. I found through wikipedia that photosynthesis has a linear increase in effectiveness from 0 to about 100 watts/meter^2 , and is flat after that (and in fact decreases at some level).

I am considering a settlement at 50 degrees north, 90 degrees west (0 degrees longitude defined to be the longitude exactly away from the sun at the spring equinox).

After orbit simulation, taking a sample of solar power at the equator and at 50 degrees north yields an interesting plot, y axis is in kilowatts per meter^2 :

Solar irradiance at top of atmosphere and intensity at the surface at two points 280 days past the spring equinox

Even though daily integrated solar power is dramatically lower at 50 degrees north (1.13 megajoules in the day) vs the equator (19.3 megajoules in the day, per square meter), energy available to photosynthesis is not as different, still 1.13 megajoules at 50N vs about 3.9 megajoules at the equator (once again, per meter^2) and not as far below the maximum usable power of 100 W m^-2 .

This surprised me. There are a lot of other factors in the growing season, such as temperature, but in terms of energy available it doesn't seem like the growing season would be dramatically reduced from the Earth, and in fact given the slightly longer year (just over 405 24-hour days) may actually be a bit longer than the equivalent latitude on Earth.

But, it's not quite as simple as that, I'm sure. So the question is: given the data that I have available and can compute, can I approximate a growing season? If so, how? Is there data on how much energy earth crops need to productively grow per plant, and how much photosynthesizing leaf area each plant has? I presume leaves will be at random angles to the sun and some will shadow others and I'll have to work out a model for that, but right now I'm just trying to look at it from an energy availability point of view. Thanks!

For the curious, solar energy through the year at the two points. Each of the two points has two curves, one for irradiation at the top of the atmosphere and the other corrected for air mass and presumed to be at the surface. Time should be advancing at about 24 days per second, or one day per frame:

https://i.imgur.com/EjpIFky.gifv

And amending the question with a little more information, here is a map of the seasons over time and the distance of Home from its primary in the lower right panel:

Axial deflection from 90 degrees, velocity of deflection, and distance from primary

Additional edit:

It does seem that temperature and other climate factors are probably more important than availability of sunlight. Given that, is there a simplified approach to modeling the climate (even if in general terms) or is it necessary to attempt to adapt a GCM to this world?

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    $\begingroup$ From my own experience, the energy available from sunlight isn't a direct factor in the growing season. For example, around here many things grow better when they are shaded for part of the day. The prime factor is temperature, and the expected variation. That's why in many gardening books, you see references to the last frost date: epod.usra.edu/blog/2002/03/last-frost-dates-1.html Temperatures in turn will depend on elevation, proximity to mountains & bodies of water, precipitation (desert areas tend to have much greater daily variations), $\endgroup$ – jamesqf Mar 1 '18 at 23:49
  • $\begingroup$ @jamesqf, yep. Living on the north shore of Lake Superior at the top of a large hill, depending on where I plant seeds I have four different zones for plants in a steeply sloped half acre yard. It comes down to wind breaks, shade, elevation, and even the type of soil. $\endgroup$ – Dan Clarke Mar 2 '18 at 0:53
  • $\begingroup$ I'm curious, how did you arrive at the graph of solar power? You said you simulate the orbit. Could you elaborate? I've been looking for good tools to use, but so far I do it all by spreadsheet. $\endgroup$ – n_bandit Mar 9 '18 at 20:19
  • $\begingroup$ @n_bandit Yes, I do an n-body simulation of the system). From that I know the positions of the center of mass of the sun and planets. Then I apply rotation to the planet in question so I know given the center of mass how the surface is oriented. I select a surface point that I'm interested in and know the vector to the sun, then do a change of basis to get the vector to the sun relative to the surface, and from that I can find insolation (and now I modulate that insolation by atmosphere depth from that angle as well to give insolation at the surface). $\endgroup$ – sudnadja Mar 9 '18 at 21:11
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I don't think any of the information you give can be used to predict a temperature range on planet home, let alone model a cycle of agriculturally viable climates.

Maybe my logic is flawed, but I think the fact that we struggle to explain the differentials in climate here on Earth should be an indicator of this. Causally linking changes in the amount of solar irradiation (of which I believe there were none) to climate change is a challenge.

Extrapolation of the climate on a planet from the astronomical arrangement of its solar system would appear to be impossible, and even if it weren't, very low confidence.

You did say the planet is very similar to Earth, in some ways. This could narrow down the temperature range down to a few thousand degrees, if given some very specific constraints, and a lot of more information.

Maybe an indicator of this would be the fact that the recognizable cycle of farming seasons we have now has only been for a few thousand years old. And it's very possible the very advent of agriculture has affected the climate that made it possible. There have been ice ages on Earth.

I guess you could say the planet was in fact, identical to Earth, and start with initial conditions based on the Earth today, and then create a model based on the geology of our Earth, but I feel like at that point, there's no real reason to do this, and it would seem very contrived.

What you could do instead is conjure up a planet with a climate (or a cycling pattern of climates) and then give an explanation based on the information you've given and the geological features: working backwards, essentially, choosing a model that fits the data that is convenient for you.

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