Orbital Calculations?

[Cray]

For another setting, I was sketching out a star system with 0.5 solar mass primary and a planet orbiting it with a 30,000,000-kilometer semi-major axis.

It's easy enough to find the year length (46 Earth days), but I had a few questions:

1. Given an eccentricity of 0.5, how many days does the planet spend in each quarter of its orbit (quarter: as measured by distance along the circumference)? Basically, how long is each season? Also, what is the periapsis in kilometers?

2. Given an eccentricity of 0.7, how many days does the planet spend in each quarter of its orbit?  Also, what is the periapsis in kilometers?

Assuming for sake of argument that the planet was habitable and Earth-like, and that it maintained a fairly short rotation (under 48 hours), what would that high eccentricity and short year do to weather?

The short heat pulse and long cooling are predictable, but they're happening over (basically) a long month. What sort of winds and weather patterns would develop? General chaos, or something regular and predictable?

[stevebowers]

The perigee is easiest; with an eccentricity of 0.5, the periastron is 15,000,000km. Half the semimajor axis.
With an eccentricity of 0.7, the periastron is 9,000,000 km.

Summer in the 0.5 case would be short and hot, with violent weather especially just after the periastron. Winters would be cold, especially after apastron.

Summer would be very extreme in the 0.7 case, but short; winter would be long and cold. I think you'd need weather machines in this case, just to make the planet even slightly habitable.

I haven't worked out how long each quarter would last yet.

[stevebowers]

I don't know how to work out the lengths of the quarters using maths, so I plugged these into Celestia and estimated it. I chose the periapses as the start of the orbits.

For a planet with a 46 day year and an eccentricity of 0.5, the two shorter (summer) quarters would each last about 8 days each. The two longer (winter) quarters would last about 15 days each.

For a planet with a 46 day year and an eccentricity of 0.7, the two shorter (summer) quarters would each last about 7 days each. The two longer (winter) quarters would last about 16 days each.

[Cray]

Thank you, Steve. The research is much appreciated.