White Dwarfs and Trojan Planets

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White Dwarfs and Trojan Planets

Post by SoulStorm on Sat May 28, 2011 7:27 am

Hi All,

I'm working on a science fiction novel with a cosmology that I'd like to be as realistic as possible. Given that my knowledge of physics and astronomy is just enough to get me in trouble, I'm hoping for some help from a few of the brilliant minds who frequent this site.

I'd like the setting to be a roughly earth sized trojan planet rotating around a white dwarf at ~ .01 AU. If my calculations of the Roche Limit are correct, I can use a gas giant almost as large as Jupiter as the secondary mass.

Roughly how large would the gas giant appear to be from the trojan planet?

My understanding is that a planet can stay within the habitable zone of a white dwarf for 3-8 billion years.

I understand that due to the planet's proximity to the star, magnetic field drag could potentially cause the planet's orbit to decay. Is it plausible to suggest that being located at a trojan point could prevent such decay?

I also understand that a planet's conductivity has an effect on magnetic field drag, but I don't understand the direction of the relationship, so some help there would be nice.

I'd like to exploit the magnetic field of the white dwarf by creating a persistent geomagnetic storm around the trojan planet to create a planet wide aurora effect. I'm picturing something like what happened during the Carrington Event of 1859, though a bit stronger if possible. Basically, I'd like the night sky to be lit with swirling colors bright enough to create a perpetual twilight. Is this possible assuming the magnetosphere of the trojan planet is strong enough? What would be the long term consequences of such a dynamic?

Could being a trojan planet prevent tidal locking with the white dwarf? If not, is there another way to create a stable spin-orbit resonance other than 1:1 besides messing around with the eccentricity of the orbit? Basically, I'd like the planet to have a day/night cycle that doesn't decay over time.

How would the tidal forces of the white dwarf and gas giant affect the trojan planet?

One last question, if you were a technologically advanced race that wanted to keep your star burning hot enough to maintain the habitable zone of your planet, how would you do it?

Right now I'm contemplating a device that converts pressure into heat. Science fiction or science fantasy?

Thanks for the help!

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Re: White Dwarfs and Trojan Planets

Post by Sedna on Sat May 28, 2011 12:30 pm

I'm not the expert here, but I can answer a few of your questions.

SoulStorm wrote:I'd like the setting to be a roughly earth sized trojan planet rotating around a white dwarf at ~ .01 AU. If my calculations of the Roche Limit are correct, I can use a gas giant almost as large as Jupiter as the secondary mass.

I suppose that by "white dwarf" you mean that your planets orbit a remnant of a Sun-like star. But there's one big problem: a planet orbiting closer than 1 AU is (most likely) engulfed by its parent star during the red giant phase. So your planets at 0.01 AU have no chance to survive.

SoulStorm wrote:Roughly how large would the gas giant appear to be from the trojan planet?

There's one formula that can help you: delta = 2*arcsine*(r/d), where delta is the apparent diameter of the gas giant (in arcseconds), r is the distance between the gas giant and the trojan planet (in km), and d is the radius of the gas giant (in km).

SoulStorm wrote:My understanding is that a planet can stay within the habitable zone of a white dwarf for 3-8 billion years.

I'm totally not sure about this, it's hard to imagine an HZ so close to a white dwarf.

SoulStorm wrote:Could being a trojan planet prevent tidal locking with the white dwarf?

As your trojan planet is outside the gas giant's hill sphere, it will be tidally locked to the white dwarf. And there's simply nothing you can do to avoid it, ot create a hard sci-fi story...

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Re: White Dwarfs and Trojan Planets

Post by Sirius_Alpha on Sat May 28, 2011 1:21 pm

Welcome Smile

SoulStorm wrote:Roughly how large would the gas giant appear to be from the trojan planet?
You'll need the radius of the planet, but it should be fairly easy to solve for.



Of course once you get your θ, you'll have to double it since the planet's true size is twice the radius.
Edit: I see Sedna beat me to it. Razz

Soulstorm wrote:I'd like to exploit the magnetic field of the white dwarf by creating a persistent geomagnetic storm around the trojan planet to create a planet wide aurora effect.
Remember aurorae do not come from magnetic fields, but from particles becomming entrapped in a planetary magnetic field and funnelled to the magnetic poles of the planet. So what the star needs is less a magnetic field, but rather, a decent amount of outflow of charged particles. White dwarfs do not satisfy this criteria.

SoulStorm wrote:Could being a trojan planet prevent tidal locking with the white dwarf?
No.

SoulStorm wrote:If not, is there another way to create a stable spin-orbit resonance other than 1:1 besides messing around with the eccentricity of the orbit? Basically, I'd like the planet to have a day/night cycle that doesn't decay over time.
You could perhaps put enormous thrusters along the equator of the planet to keep the thing spinning.

SoulStorm wrote:How would the tidal forces of the white dwarf and gas giant affect the trojan planet?
The white dwarf would lock the planet into a 1:1 spin-orbit resonance. Perturbations from the gas giant may cause it to wander slightly from that, with the everpresent tidal influence from the star damping the effect, similar to what happens at Io.

SoulStorm wrote:One last question, if you were a technologically advanced race that wanted to keep your star burning hot enough to maintain the habitable zone of your planet, how would you do it?
I'd probably move to a red dwarf, hoard gas giants from neighboring systems, and drop them in as necessary to keep the fire going. That should keep you happy for quadrillions of years or if you keep it supplied.

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Re: White Dwarfs and Trojan Planets

Post by Sirius_Alpha on Sat May 28, 2011 1:28 pm

Sedna wrote:delta = 2*arcsine*(r/d), where delta is the apparent diameter of the gas giant (in arcseconds), r is the distance between the gas giant and the trojan planet (in km), and d is the radius of the gas giant (in km)
If you use arcsin, you're basing the distance between the planet and the trojan off the leg of the triangle between the centre of the trojan and the limb of the gas giant (the hypotenuse). It's a bit skewed mathematically, but for the application we're considering here, the difference won't physically translate to much.


Last edited by Sirius_Alpha on Sat May 28, 2011 1:28 pm; edited 1 time in total (Reason for editing : Wording)

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Re: White Dwarfs and Trojan Planets

Post by SoulStorm on Sat May 28, 2011 2:03 pm

Thanks for the data Sedna and Sirius Alpha!

You may find the following journal article regarding white dwarf stars interesting Sedna:

http://arxiv.org/abs/1103.2791

I know I've read at least one article about a planet being found within the habitable zone of a white dwarf. I'll see if I can find it for you when I have a chance.

There's also some good discussion about how a planet could end up within the habitable zone of a white dwarf here: https://analogsf.com/aspnet_forum/messages.aspx?TopicID=1342

Pity about being tidally locked. I wish there were a way around it that doesn't involve the use of technology. My understanding is that Mercury escapes being tidally locked due to the eccentricity of its orbit. However, if I understand things correctly, orbits become more circular over time, though I don't know the process which causes that to happen.

Still, being tidally locked isn't a deal breaker as studies have shown that an atmosphere even 1/10 as thick as that of earth's would probably circulate the warm air to the dark side of the planet. Not my preference, as I expect this would still limit the habitable areas of the planet, but it could work. I could probably even use a super earth to increase the habitable surface area.

Abandoning the idea of using a white dwarf for a moment, if you wanted to create the persistent aurora effect on a habitable planet how would you do it? Bonus points if the planet has a day/night cycle. Smile

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Re: White Dwarfs and Trojan Planets

Post by SoulStorm on Sat May 28, 2011 2:19 pm

Sirius, I knew that the magnetic field interaction in and of itself wouldn't create the aurora effect I was looking to achieve. However, I was hoping that if the planet was within the magnetic field of the white dwarf, that the electrons and protons of the white dwarf would propogate along those lines of force and interact with the charged particles in the trojan planet's atmosphere. What I didn't know is if the planet was close enough to the sun's magnetic field for this to happen. Perhaps my original question wasn't clear enough? Or would this just not work?

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Re: White Dwarfs and Trojan Planets

Post by Sirius_Alpha on Sat May 28, 2011 2:27 pm

SoulStorm wrote:My understanding is that Mercury escapes being tidally locked due to the eccentricity of its orbit. However, if I understand things correctly, orbits become more circular over time, though I don't know the process which causes that to happen.
As a planet swings around at perihelion, orbital energy is lost to tidal heating.

Tides will slow a planet's rotation rate down until, at perihelion, the rotation rate is synchronous. For a circular orbit, there's no defined perihelion, and the planet rotates synchronously throughout the orbit. Consider two rates: the rotation rate (the amount of degrees the planet rotates in some unit of time) and the orbital rate (the amount of the orbit that the orbit the planet traverses in some unit of time). For a circular orbit, the orbital rate and rotation rate will be constant. For an eccentric orbit, the rotation rate will be constant over the scale of an orbit, but the orbital rate will vary (Kepler's laws of orbital motion -- planets near perihelion orbit faster than when they are at aphelion).

There are "fluid bodies" (like gas giants), whose density distributions don't vary terribly much, and "rigid bodies" like terrestrial planets. Let's consider the case of a fluid body first.

Tides on the fluid body will slow the rotation rate until it matches the orbital rate at perihelion (where tides are strongest). For a circular orbit, this is of course synchronous, but for an eccentric orbit, since the orbital rate changes over the course of the orbit, no true synchronous rotation state is possible. So for the brief moment at perihelion only, tides will force the planet to rotate synchronously with its star. An observer "on" the planet will see the star stop moving in the sky at perihelion, then continue on its way after perihelion. This state of being in synchronous rotation at only perihelion due to an eccentric orbit is called "pseudosynchronous rotation."

Rigid bodies have mass concentrations which can affect their end tidal state. Typically, the long axis of the body will want to point toward the primary (it's a bit more complicated than that, but I'll not elabourate much here). This natural shape deformation can be more pronounced than that of the tidal deformation, and thus dominate the tidal evolution of the rotation rate. The rotation rate will slow until the long axis of the body points to the primary during perihelion.

Now in the case of a spin-orbit resonance, Mercury has gotten itself stuck in a 3:2 spin-orbit resonance. Each time it's at perihelion, the same axis is aligned with the sun (albeit, opposite sides of the planet each time). As a result, Mercury has reached its final tidal state. As such, there is no net tidal influence on Mercury's rotation rate, and therefore, minimal tidal circularisation of its orbit.

SoulStorm wrote:Abandoning the idea of using a white dwarf for a moment, if you wanted to create the persistent aurora effect on a habitable planet how would you do it? Bonus points if the planet has a day/night cycle.
I'd have the planet the orbit a highly active star.

SoulStorm wrote:I was hoping that if the planet was within the magnetic field of the white dwarf, that the electrons and protons of the white dwarf would propogate along those lines of force and interact with the charged particles in the trojan planet's atmosphere. What I didn't know is if the planet was close enough to the sun's magnetic field for this to happen. Perhaps my original question wasn't clear enough? Or would this just not work?

One of the reasons white dwarfs last so long is because they don't put out a lot of energy. They aren't fusing, either, so there's nothing to encourage a source of particles away from the white dwarf. These stars aren't even able to keep molecular gas from accreting onto their surface.

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Re: White Dwarfs and Trojan Planets

Post by SoulStorm on Sun May 29, 2011 3:00 am

The following quote is from the article, "Planets Around White Dwarfs" written by Jianki Li et al.

"If a white dwarf is magnetic, then, under certain circumstances, a current system may develop linking the white dwarf and the planet, much like in the Io-Jupiter interaction."

Since the interaction between Jupiter and Io does create an aurora effect, it looks like a magnetic white dwarf may at least provide a semblance of the effect I'm trying to achieve assuming appropriate atmospheric conditions and a significant iron core. However, I'm open to the possibility that I'm still missing something.

The main example seems to presume a planet with an orbit of 200 white dwarf radii. In such a case, the magnetic drag is presumed to cause the planet to crash into the white dwarf in ~100,000 years.

A trojan planet orbiting a 6,000 km radius white dwarf at ~.01 AU would have an orbit of ~250 white dwarf radii. Presuming a magnetic field ranging between 10^7 and 10^9 G^3, the trojan planet should experience a magnetic field somewhere between 0.64 and 64 Gauss. All the math in this paragraph is mine, so it may be wrong. Smile The strength of the white dwarf's magnetic field may also chip away at the planet's atmosphere, but I don't know that for certain, or how significant the impact on the atmospere would be if it did.

I'm also still wondering if the Gas Giant could prevent the trojan planet's orbital decay if the Gas Giant's conductivity were low enough to prevent it's own orbit from being affected by magnetic field drag.

Thanks for the lesson on Tides Sirius. It's going to take me some time to digest, but I'll work on it.

In the meantime, I'm wondering why a planet orbiting a white dwarf can't have a 3:2 spin-orbit resonance like mercury has with the sun. If the answer is in what you've already told me, just indicate so and I'll study harder. Smile

Regarding having the planet orbit a highly active star, I've looked at that possibility but I could never figure out how to make it work. Red dwarfs actually sound like they're so energetic for the first 2-3 billions years that they could strip away a planet's atmosphere. I've also looked at tidally locked binary red dwarfs, but that seems to create very unstable systems. I even looked at pulsars, but realized very quickly how crazy that idea was. My examination of yellow dwarfs didn't yield results either. I also looked at earth sized moons to possibly get an Io-Jupiter sort of dynamic going, but then the Roche limit became a problem. So trust me, I've tried.

All the above stated, if I've learned anything during my research, it's that new discoveries are being made on a daily basis. If the answer I'm looking for isn't out there right now, it probably will be soon.

Right now, however, white dwarfs are the closest thing I've found to a working solution. As long as I can reach a point of plausibility, I think I'll be satisfied.

Thanks again for the help!




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Re: White Dwarfs and Trojan Planets

Post by Sedna on Sun May 29, 2011 5:05 am

Sirius_Alpha wrote:If you use arcsin, you're basing the distance between the planet and the trojan off the leg of the triangle between the centre of the trojan and the limb of the gas giant (the hypotenuse). It's a bit skewed mathematically, but for the application we're considering here, the difference won't physically translate to much.

That's right. It just that when I use arcsine, arccos, arctan, I always end up thinking sine, cosine, tangent. And at very small angles (like this case), sines and cosines are (almost) equal.

SoulStorm wrote:You may find the following journal article regarding white dwarf stars interesting Sedna:

http://arxiv.org/abs/1103.2791

Indeed, that's very interesting, I've never thought about that. Thanks for the link !

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Re: White Dwarfs and Trojan Planets

Post by Sirius_Alpha on Sun May 29, 2011 10:30 am

SoulStorm wrote:Since the interaction between Jupiter and Io does create an aurora effect, it looks like a magnetic white dwarf may at least provide a semblance of the effect I'm trying to achieve assuming appropriate atmospheric conditions and a significant iron core. However, I'm open to the possibility that I'm still missing something.
The Jupiter-Io interaction produces aurorae on Jupiter. Remember what causes aurorae (even at Jupiter): Charged particles are funnelled down to the magnetic poles where they interact with an atmosphere. If you want this white-dwarf-orbiting planet to have aurorae, then the planet needs to have a source of charged particles. A magnetic field alone isn't going to give you aurorae.

SoulStorm wrote:I'm wondering why a planet orbiting a white dwarf can't have a 3:2 spin-orbit resonance like mercury has with the sun.
It can, but you mentioned you didn't want to have to give the planet an eccentric orbit.

SoulStorm wrote:Red dwarfs actually sound like they're so energetic for the first 2-3 billions years that they could strip away a planet's atmosphere

Two papers from this thread:
Atmospheric mass loss by stellar wind from planets around main sequence M stars
The Effect of a Strong Stellar Flare on the Atmospheric Chemistry of an Earth-like Planet Orbiting an M dwarf
Moreover, one of the transiting planets that have been found is a super-Earth with a thick atmosphere in a close orbit around a red dwarf (Gliese 1214 b).

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Re: White Dwarfs and Trojan Planets

Post by SoulStorm on Sun May 29, 2011 12:15 pm

Sirius_Alpha wrote:If you want this white-dwarf-orbiting planet to have aurorae, then the planet needs to have a source of charged particles. A magnetic field alone isn't going to give you aurorae.

Sorry, I should have been more clear. That's what I meant when I said appropriate atmospheric conditions. I know that Earth's aurora is mostly due to oxygen and hydrogen ions and that the bright blue aurora at Io's poles is thought to result from Io's vulvcanism along with Io's electrical connection with Jupiter. Sodium ions are also thought to play a significant role in Io's aurora if memory serves.

Though I'm fine with a certain amount of vulcanism on my trojan planet, I don't want to go overboard. I'm still working on atmospheric and geological features. I know that I want a lot of mountains and valleys on land, but not so many mountains that I critically hinder advection. I'd also like an atmosphere that produces a lot of storms, so I'm also going to need a lot of surface water, perhaps more than is on Earth, but I'm still working on that. The electrical connection with the white dwarf should create some striking electrical phenomena (no pun intended).

Sirius_Alpha wrote:It can [have a 3:2 spin-orbit resonance], but you mentioned you didn't want to have to give the planet an eccentric orbit.

Yep, I totally borked that up, you got me there. Embarassed I didn't understand that Mercury's 3:2 spin-orbit resonance was stable. I just presumed that Mercury's orbit would become circular eventually and end up tidally locked with the sun. You cleared that misunderstanding up for me in your last post. Many thanks for that! Very Happy This does bring to mind a few more questions, however.

Can the planet be a trojan and still have a 3:2 spin-orbit resonance?

Can the planet have the eccentricity necessary to achieve a 3:2 spin-orbit resonance while still remaining within the dwarf star's habitable zone (or at least close enough not to cause habitability problems)?

Is it possible to use these dynamics to prevent the planet from crashing into the star?

I'll check those papers out Sirius. Thanks again, you've been incredibly helpful!

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Re: White Dwarfs and Trojan Planets

Post by Sirius_Alpha on Sun May 29, 2011 1:34 pm

SoulStorm wrote:Can the planet be a trojan and still have a 3:2 spin-orbit resonance?
I can't think of a reason it couldn't.

SoulStorm wrote:Can the planet have the eccentricity necessary to achieve a 3:2 spin-orbit resonance while still remaining within the dwarf star's habitable zone (or at least close enough not to cause habitability problems)?
Probably. The eccentricity you would need for the 3:2 spin-orbit resonance could dominate the seasonal cycle of the planet though. A Mercury-like eccentricity shouldn't pose too strong an issue to habitability though given the planet's oceans acting as a sort of heat sink. And of course given the very short orbital period of this planet, thermal inertia is going to work in your favour. The "winters" and "summers" will be but a few hours, so it's not like there's really enough time to devastate the ecology.

SoulStorm wrote:Is it possible to use these dynamics to prevent the planet from crashing into the star?
The trojan configuration with the gas giant probably adds some stability, as opposed to both planets orbiting the star at roughly the same distance without a resonant configuration. Having the gas giant completely absent would be a great boost to stability.

It takes a lot of orbital energy to be lost to crash into a star. In our Solar system, it is literally easier to get to Mercury than it is to get to Pluto from an orbital energy perspective. Climbing up or down a gravty well takes energy, and the sun rests at the bottom of a very deep one. It's not an exaggeration to say we lack rockets that can reach the sun on their own.

The case of the white dwarf is a more extreme one given the star's smaller size -- you have to climb much further down the gravity well to reach the surface.

I would be more concerned about the planet's Roche limit. You can't get too close to the star before the stellar tides begin to disrupt the planet. It depends on the planet, the mass of the star, and the luminosity of the star, but the Roche limit isn't terribly far from the habitable zone. Astronomers frequently detect stars with disks and surface pollution due to asteroids that have come too close and were tidally disrupted.

SoulStorm wrote:Thanks again, you've been incredibly helpful!
Any time Very Happy

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Re: White Dwarfs and Trojan Planets

Post by SoulStorm on Sun May 29, 2011 10:46 pm

You made my day Sirius! It's always nice when things work out! cheers

Sirius_Alpha wrote:Having the gas giant completely absent would be a great boost to stability.

I'm just going to paste some text from the journal article, "Planets Around White Dwarfs" written by Jianki Li et al.

I didn't understand everything from the article, and I don't want to risk conveying incorrect information.

"Using the parameters adopted, an Earth-type core at twice the solar radius would produce a magnetic heating luminosity of 10^29 erg s^-1. For such an orbit, the orbital period is 10.2 hours. The inward drift velocity can be calculated by (7) for a given heating rate. For an orbital period of 10.2 hour, and a heating rate 10^29 erg s^-1 the predicted value is...13.3km yr^-1  
The characteristic time for the planet to coalesce with the surface of the WD
is therefore 10^5 years."

The article goes on to state, "For orbital separations significantly greater than 200 Rw the white dwarf will...not be subject to signicant heating and the planet could have a prolonged lifetime. In contrast, if the separation is significantly smaller than 200 Rw the heating is so large that the planet will merge with the WD in a time scale much shorter than 10^5 yr.  In general for a system with 1000 Rw < 10 Solar Radii, a merger would occur in less than a Hubble time."

The rough draft of my white dwarf has a 6000 km radius, and the trojan planet orbits at ~.01 AU. Now that we're using an eccentric orbit, I can probably push the orbit out a bit further. I'm not married to these numbers, I just wanted to get a rough idea of how the numbers would crunch. These numbers put the trojan at 250 Rw.

There is also some interesting data at the end of the article, "Our model has been based on specific and possibly restrictive assumptions. There are two possible variants of the model that may allow the magnetically interacting planet to have a significantly longer lifetime. The degree of ablation of the mantle material of an Earth-type planet is not certain. If the ablation is incomplete, the effective resistivity of the “core”—a core covered by a mantle shell—is larger than that of the atmosphere of the WD. The heating would then be in the “core” of the planet, reminiscent of the Io-Jupiter system, and this could be the situation in the early phases of the magnetic interaction. In this phase, the electrical current is small, and so the characteristic timescale for the planet to drift inward could be much longer than 10^5 yr. The heating of the planet “core” would accelerate the evaporation of the surface layer of the mantle material until eventually the heating would occur in the WD atmosphere, in the manner outlined in this Letter. Another possibility is that the parent planetary system may be quite unlike our own solar system. If the conducting core can be more massive than we have assumed (e.g., from a Jupiter-like planet), the inward drift time would also be significantly increased."

There are a number of variables I can play around with such as the magnetic fields of the 2 bodies, orbital distance, the conductivity of the planet, core size, mantle composition, etc. Modifying these variables can slow the inward drift, but the only way I can think of to stop it is to use a trojan planet in orbit with a gas giant (which shouldn't have enough conductivity to form a circuit with the star). Of course, the gas giant's moons could be a different story, so that may introduce some interesting dynamics.

I'll probably just hand wave most of it, since this isn't going to be a hard sci-fi novel. I just want an authentic setting that is consistent with the traits of the aliens living on the planet. I just need plausibility so that an informed reader can suspend disbelief. Plus, it's kind of fun to consider the possibility of there being a planet out there somewhere like the one you've imagined and brought to life in prose. Very Happy

I don't need the circuit between the white dwarf and the trojan planet to be so strong that it burns a hole to the core of the planet. I just need it to be strong enough to give me the aurora effect and some crazy lightning phenomena around the poles. I'd like to think Tesla would be happy with the direction this novel is heading in. Very Happy

As always, thanks Sirius!

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Re: White Dwarfs and Trojan Planets

Post by Lazarus on Wed Jun 01, 2011 12:05 pm

One question would be how to get the jovian and the terrestrial planet into such a close orbit in the first place.

Presumably you could get terrestrial planets there by second-generation planet formation from a fallback disc. Not sure how viable this mechanism would be, certainly there are no known examples of such planets around a white dwarf, though presumably the PSR B1257+12 planets formed from supernova fallback material.

Whether such discs would be massive enough to form a jovian and migrate it into the close orbit is another matter. You could end up with a first-generation jovian in close orbit if it is engulfed by the red giant and ejects the envelope as it migrates inwards (I'd guess you'd then end up with a sdB star + a jovian in close orbit, the sdB star would eventually evolve into a WD). However such a system does not look like a good bet for forming Trojan terrestrial planets, presumably any remnant disc would lie outside the orbit of the jovian and be truncated at its inner edge by the planet, it's not entirely clear to me how you'd get a terrestrial planet into the Trojan points from this configuration.

(Plus you then have to worry about tidal dissipation and what this would do to the Trojan...)
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