How can I find out the length of seasons and solar year on this planet?

Question

This planet is fictional, but I want it to be credible all the same. It is the same size and composition of Earth, orbiting a sun-like star. But that's where the similarities end.

The planet's orbit is far more elliptical, coming just into the 'too hot' section for a few days twice on its closest approach. With a 'spring time' like season between these two 'burns'. On the other side of its orbit, it is outside in the 'too cold' region for about short spell (I'd prefer no less than a month, but the math is more important than my preferences).

(above is a series of orbits. description is most similar to orbit D)

The question is then, within these constraints, how do I figure out how long each 'season' is? And would the planet's speed be constant throughout its journey around the sun?

EDIT:

The parent star is exactly like our sun. Same age, size, and composition. The planet I am talking about has the same composition as Earth, and the same atmosphere. as for the orbit. I figure that the star is in the centre of Focal Point 1. With the closest approach being 0.49 AU and the furthest distance being 3.2 AU (if wikipedia is correct about our sun's Circumstellar Habitable Zone being between 0.5 and 3 AU. If not, please tell me and I'll correct this.)

As for the planet, it has the same axial tilt as Earth, and the same wobble. Sorry if this is a bit boring to those that can do the actual math involved, but my focus is more on psychology, sociology, and biology. This is slightly above my paygrade.

As a bonus question. How long would the surface be uninhabitable during the two closest approaches? Are we talking just the period the planet is in the 'too hot zone'? I could imagine it would become unbearably hot well outside the 'too hot zone', making the dual summers too much for the populous to bear.

Answer 1

Orbit "D" is physically impossible. You can't have the planet passing into the "too hot" zone twice in a single year. (Explanation as suggested in the comments: This is because the sun is always at one focus of the ellipse, and the two ends of an ellipse are the closest and farthest points from the focii. Ergo, when the planet passes the end of the ellipse near the sun, it must be closer than it is anywhere else, so you can't have two sections on either side passing through the "too hot zone", closer to the sun, with the end point farther away in the habitable zone.)

If it is really important to have two "burns" separated by a springtime in one direction and a long winter in the other, you'll need a more complicated system. Exactly what that would have to look like, I'm not sure. I can't think of any plausible arrangement of suns that would give that effect on a consistent basis, as opposed to say, once every century.

If you're OK with just one hot spike per year, the length of the year is easy to calculate. It's just $2\pi\sqrt{\frac{a^3}{GM}}$, where $a$ is the semimajor axis (the average of the closest and farthest distances of the planet from the star). Since the star is exactly the same as our sun, we can simplify this to get the length in terms of Earth years: $(\frac{r_p+r_a}{2})^\frac{3}{2}$, where $r_p$ (for perihelion) is the smallest distance between the planet and sun, measured in AUs, and $r_a$ (for aphelion) is the largest, also measured in AUs. For your stated values of 0.49 and 3.2 AUs, the year on this planet will be approximately 2.5 Earth years.

Figuring out just how long each season is, and how long the world would be uninhabitable, is significantly more complicated, because the planet's speed is not constant through the year. It moves much faster when it is closer to the sun, so the summer will be much shorter than the winter. Additionally, the hottest period will not be centered on the time of closest approach to the sun, nor will the coldest period be centered on the time of farthest distance, due to thermal inertia (the time it takes for the mass of the planet to warm up or cool down). Rather, the hottest part of the year will be shifted later than the closest approach to the sun.

If I were you, I'd avoid stating specific details of the orbit in any story using this world. Go ahead and figure out how long the year needs to be as a whole, and then just decide how long you need the "too hot" and "too cold" periods to be, and accept that there probably is a solution for $r_p$ and $r_a$ that will give you those values even if you don't know exactly what it is. And if there turns out not to be one, well, you've already established that this planet is only inhabited due to divine intervention anyway, so that's just one more thing that the gods have employed their powers to tweak, through solar shades or manipulating the planet's emissivity or whatever.

Answer 2

https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

You've made a number of errors in your set up.

Some are really surprising - such as your comment that the planet's axis[sic] can be pointing away from the star. (hint: two axes, if one points "away", the other will be pointing "towards").

Ellipse D doesn't appear (to me) to have the star at one focal point.
The Habitable Zone is determined by using a wide range of atmospheric pressures; using 0.5 & 3 AU for your two circles is really extreme. Earth wouldn't be habitable in either Venus' or Mars' orbits (0.7 & 1.5 AU), going to 3.2 AU is over-kill, imho.

As far as the computation of the length of the year and the seasons, the year is simple math, given the orbit.

The seasons are a lot more problematical. First, if Summer is fatal (for surface life) then wouldn't there be two "shoulder" seasons where it's just really really hot (but not fatal)? Similarly with a winter where (say) the atmosphere freezes out (or whatever). The number of seasons depends. Given the "thermal latency" that our oceans give our planet, I don't know if atmospheric freeze-out is credible for short periods (only months-long) of low insulation. I'd say that if I don't know then most people won't either (I am not claiming to be very knowledgeable in this, just significantly more than the "average" adult - but much less than those studying these issues as well as those arm-chair experts whose opinion you should also get.)

Anyway, given the interactions between the air and water and winds and surface, it is really difficult to predict what the seasons would look like even if we just moved Earth into Venus' or Mars' orbits (and didn't change spin or tilt). So, I say you're free to draw a smooth periodic function (see https://en.wikipedia.org/wiki/Periodic_function) of air temperature vs time. (by smooth I mean no "sharp" corners, everything happening over weeks or months).

A circle is an ellipse (with positions F1=F2). At the other extreme, consider a parabolic orbit. Toss a rock up and it is at its fastest leaving your hand slows to a (vertical) stop and plunges downwards (and in the absence of air resistance) with maximum speed at impact. This is similar to a highly elliptical orbit. The planet will be traveling its slowest as it passes through its apogee, and fastest as it passes thru its perigee.

I think you need to break down your question into a lot of smaller pieces. It is straight-foward math to determine what the length of a day is (which is most of what determines the seasons here) given the axial tilt of the planet and its spin. It seems to me, I recall that the difference between Summer and Winter due to the distance to the Sun is (estimated at) about 0.5°C, but don't hold me to that, please. I think what I would do is compute the days' lengths and apply them to the two limiting circles (inner and outer) to determine seasons (see diagram near wikipedia Ellipse Parametric_representation). and then interpolate between them.