TL;DR

To make a long story short, the planet will lose its atmosphere and some of its crust, but most of it will remain intact, even in the worst-case scenario. That should be the major effect you'd have to deal with.

Nebula properties

Let's review some key characteristics of planetary nebulae:

• Density: $\sim10^2$ to $\sim10^6$ particles/cm$^3$
• Expansion rate: $\sim10$ km/s
• Temperature: $\sim10000$ K

The central stars are extremely hot (20,000 to 200,000 K) and luminous; their peak emission is likely at ultraviolet wavelengths - too short for the human eye to see. This also means that each photon is more energetic than a photon emitted by the Sun, on average. That's not fantastic for an ozone layer. I suspect that there would be atmospheric loss on the planet orbiting the companion star, although at a distance of 100 AU, this could be minimal.

Ablation

Stellar winds and outflows tend to have negative effects on planets in the nearby vicinity. For example, some Hot Jupiters may have their atmospheres stripped away and become chthonian planets, thanks to intense radiation from their parent stars. The same thing should happen here - and keep in mind that the wind of the central star is likely much greater than the nebula's expansion rate.

For planets close enough to the central star, they will gradually be ablated, and have their atmospheres stripped away. This is one place where I disagree with PipperChip's otherwise awesome answer. It's thought that planets with small enough orbits may even form large tails, thanks again to the ionizing radiation. In short, yes, for the relatively brief duration of the planetary nebula (tens of thousands of years, perhaps), the planet could be stripped away, depending on how close it is. It depends on the binary separation.

The planetary mass-loss rate is $$\dot{M}=1.05\times10^{-11}\left(\frac{L_*}{5000L_{\odot}}\right)^{1/2}\left(\frac{R_p}{3\times10^4\text{ km}}\right)^{3/2}\left(\frac{a}{20\text{ AU}}\right)^{-1}M_J\text{ yr}^{-1}$$ where $L_*$ is the luminosity of the star, $R_p$ is the radius of the planet, and $a$ is its semi-major axis. Here, for an Earth-like planet, $R_p=6371\text{ km}$ and $a\approx50\text{ AU}$, in the worst-case setup. Perhaps we can assume that $L_*\approx10000\text{ K}$. Therefore, we see that $\dot{M}\sim1.13\times10^{-12}M_J\text{ yr}^{-1}$, or $\dot{M}\sim3.61\times10^{-10}M_{\oplus}\text{ yr}^{-1}$. Assuming the planetary nebula disperses after about 10,000 years, an Earth-like planet will lose 0.00000361% of its mass - enough to rob it of its atmosphere and some of its crust.

Now, the planet will already have been bombarded by stellar winds during the red giant and asymptotic giant branch (AGB) phases of the companion star's life, meaning that it will have gained mass - and possibly already had its atmosphere stripped by the AGB star's strong wind. Depending on the outcome of that particular evolutionary stage, the planet will have accreted $$M_{\text{acc}}=2.62\times10^{-9}\left(\frac{R_p}{3\times10^4\text{ km}}\right)^{2}\left(\frac{a}{20\text{ AU}}\right)^{-2}\left(\frac{M_n}{M_{\odot}}\right)M_J$$ where $M_n$ is the nebula's mass. For the same $R_p$ and $a$, and assuming $M_n=M_{\odot}$ (typical for a higher-mass star), we get $M_{\text{acc}}=6.00\times10^9M_{\oplus}$ - much less, it turns out, than will be lost. Therefore, most of the mass lost will indeed involve normal matter present on the planet from before the AGB phase.

Nebula morphology

Planetary nebulae are not always uniform, but come in a variety of shapes, including spheres (Abell 39), distorted spheroids (Helix Nebula), and bipolar lobes (M2-9). Spherical symmetry, as in the case of Abell 39, might not be great, but a bipolar nebula could be a lot better for the planet. If the axis was pointed perpendicular to the orbital plane of the other star, there is a chance the companion star (and by extension the planet) would be spared the brunt of the nebula's side effects. One problem, of course, is that bipolar nebulae exhibit strong stellar winds, which energize the particles, but those would be then directed away from the orbital plane of the planet.

I was hoping to get some actual data on this, but even in the years since this question was asked, we don't have any examples of planets in planetary nebulae, and only a small number of binary systems. A few dozen planetary nebulae have been shown to harbor binary stars as of 2011, and that figure likely hasn't risen by much since then.

However, there have been plenty of efforts to model how a binary companion would affect the morphology of a planetary nebula, and it's slowly being understood. In particular, a binary star system inside a planetary nebula may lead to ionized filaments of gas, smaller-scale structures pointing away from the central star. I don't know to what extent these could be beneficial for the planet - there isn't a drastic change in the shape of the planetary nebulae. Hopefully, more examples in the future can give us a better understanding of what you nebula would look like.

posted over 9 years ago

HDE 226868‭

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