The (Alternative) Reason for the Seasons, Part 2: Variable Star

Question

Part 1

On Earth, we have seasons due to our planet's tilt. In my last question, I proposed an alternative world where the seasons were instead caused by an eccentric orbit. Now I'm going for something more exotic. Instead of axial tilt or orbital eccentricity, what if the seasons were caused by actual variations in the energy output of the planet's host star?

Details:

• The planet is earth-like and inhabited.
• The system must be stable for at least 3,000 years.
• The length of the "seasonal" cycle must be within [-50%, +100%] of one Earth year (i.e. between six and 24 months).
• The temperature difference for some point on the planet's surface between "winter" and "summer" must be at least 15^oF and no more than 40^oF.
• The cycle doesn't have to be exactly the same as Earth's, and all of the above can be taken loosely in light of that. For instance, the cycle could repeat once every four Earth years as long as there's a cold season of some description every one Earth year, and a 60^oF temperature delta is acceptable so long as it happens less often than once per hot season.

Is it possible for a real variable star system to produce seasons with these characteristics?

And if so, what kind of variable star would be best suited to producing such a world?

If not, how close can we get? Which variables are problematic?

Importantly, the planet does not need to have formed in its current location around the star, and the system does not have to be stable across astronomical timespans. I'm not concerned about whether an inhabitable planet could realistically form and evolve life in this scenario, just whether such a planet could have seasons caused by a variable star.

Answer 1

Let's look at the two extremes: $\Delta T=15^{\circ}\text{F}$ and $\Delta T=40^{\circ}\text{F}$. In the first case, I'll say $T_{\text{min}}=50^{\circ}\text{F}$ ($283.2\text{ K}$) and $T_{\text{max}}=65^{\circ}\text{F}$ ($291.5\text{ K}$). This gives us $$\frac{T_{\text{max}}}{T_{\text{min}}}=1.03$$ Let's also assume that the planet's surface temperature is well-approximated by its effective temperature; in other words, $$T\simeq T_{\text{eff}}=CL_*^{1/4}$$ where $C$ is a function that doesn't depend on $T$ or $L_*$. Let's say that $L_{*,\text{min}}$ and $L_{*,\text{max}}$ are the stellar luminosities that correspond to the minimum and maximum temperatures of the planet. Doing some substitution, we find that $$\frac{L_{*,\text{max}}}{L_{*,\text{min}}}=1.13$$ which is actually quite small for variable stars. Now, let's convert this to a change in absolute magnitude, $m$. We have $$\Delta m=-2.5\log_{10}\left(\frac{L_{*,\text{max}}}{L_{*,\text{min}}}\right)$$ which gives us $\Delta m=-.133$ magnitudes (remember, brighter objects have smaller magnitudes).

In the second case, say $T_{\text{min}}=35^{\circ}\text{F}$ ($274.8\text{ K}$) and $T_{\text{max}}=75^{\circ}\text{F}$ ($297.0\text{ K}$). This gives us $$\frac{T_{\text{max}}}{T_{\text{min}}}=1.08$$ We then have $$\frac{L_{*,\text{max}}}{L_{*,\text{min}}}=1.63$$ and so $\Delta m=-0.53$ magnitudes.

The best kind of variable stars I can find that meet your requirements are Alpha² Canum Venaticorum variables, which have periods of up to roughly 160 days and magnitude changes of up to 0.1 magnitudes (possibly more). These stars should produce effects similar enough to what you want, within reasonable amounts.

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It is true that this type of star - massive and luminous - will not last very long, and it will be difficult - not impossible, but difficult - for planets (and life!) to form around it. However, as the last paragraph of the question states,

Importantly, the planet does not need to have formed in its current location around the star, and the system does not have to be stable across astronomical timespans. I'm not concerned about whether an inhabitable planet could realistically form and evolve life in this scenario

If anyone can find a longer-lived star that has similar pulsations, that would fantastic. For now, though, I think this is the best you'll get.

Answer 2

Variable stars that have a fixed repeating period do exist. They are known as cepheid variables and were important in astronomy as the period tells you the brightness of the star, and thus you can figure out the distance.

Look over the linked article: they are 4–20 times more massive than the Sun, and up to 100,000 times more luminous.

However, the periods only go up to about 9 months. The longer the period, the more luminous the star. I presume the luminosity varies with the mass (like normal stars), so we are looking at the largest stars, or 20 solar masses.

The other bad news is that such a star will only last for 5½ million years. So don’t expect planets to settle down and life to form in-place!