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According to (german) wikipedia, Eta Carinae has a cloud shaped like two cones with the points at the poles of the star as well as a disk around the equator. These matter clouds are mostly or only condensed stellar wind from Eta Carinae.

In my understanding, to create the dual-cone chape, the stellar wind provding the matter originates at the poles and this gives the dual cone shape (Ok?).

Why the poles? Stellar wind would, I guess, be mostly stellar matter so hot it reaches escape velocity, further pushed by the radiation. This should be isotropic. I can only guess that maybe Eta Carina has a magnetic field shaped similarly to earth with a north and south pole, nearer the equator the stellar wind is perpendicular to the field lines so the flightpaths of the charged particles are bent back inward. Near the poles the flightpaths and field lines are mostly paralell, so the magnetic field does not really act on the stellar wind. I included this guesswork on my part so you know where to start.

ETA: Turns out my guessing about stellar wind was wrong or mostly wrong, the cloud appears to be mostly shaped by eruptions in recentish history. But the question why they are shaped that way still stands.

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    $\begingroup$ The English Wikipedia talks about the conjecture on the cause of the lobes due to high rotation and gravity brightening. Not my area of expertise but looks worthy of looking at the papers. The feature is called Homunculus so look at that section in the entry. $\endgroup$ – Jeffery Thompson Sep 6 '18 at 15:05
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It seems likely due to rotation. The disklike "skirt" could be due to material that is orbiting the star that gets thrown out by the eruption, and the bipolar nature of the eruption can be because material is preferentially ejected from the poles of the star. It may seem odd that the poles, where there is less angular momentum per gram, would be better at ejecting material, but this might be due to a greater role for radiation pressure in causing the eruption. Radiation pressure produces forces that are proportional to the radiative flux, which can come out preferentially in the polar directions because this becomes the path of least resistance when rotation makes the shape of the star highly oblate.

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