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Theoretically proposed by Konstantin Batygin and Mike Brown (@Caltech), Planet Nine could have a mass of $\sim 6.3 \pm 2 M_{\oplus}$. Even though I don't know if the hypothesis is still feasible at this point, I wonder if it's possible to detect, even if trying is not worth the effort, microlensing events all across the line of its known proposed orbit, projected from our line of sight (worth regarding actual limitations considering uncertainties in orbital parameters, photometric and microlensing uncertainties, and lack of monitoring of the specific region). What do you think about it, and if it's the case, has this been done, why (not)?

Comment: maybe one challenge is to find a source behind the orbit in our epoch.

Related questions:

One hypothetical path through the sky of Planet Nine near aphelion crossing Orion west to east with about 2,000 years of motion. It is derived from that employed in the artistic conception on Brown's blog.

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    $\begingroup$ @Glorfindel I was quite surprised to see that this particular theoretical "Planet Nine" has risen to the level of a proper noun, outside of the SciFi world! :-) I want to figure out how to ask "Who decided that the proposed cause of a small group of cometary orbits (a planet) is a proper noun, despite the alternative explanation of simple sampling bias?" but I don't see how. $\endgroup$
    – uhoh
    Feb 10 at 2:27
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    $\begingroup$ Yeah, that's exactly why at first I wrote it as "planet nine" as part of the title but it got edited. Even though I honestly have the very same doubt, after a small search at least over the bibliography in Wikipedia, there are 74 reference from 2014 to 2023 that write "Planet Nine" instead of "planet nine". This could suggest that at least part of the comunity have been doing the same. What do you think @uhoh? $\endgroup$
    – nuwe
    Feb 10 at 2:59
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    $\begingroup$ I guess even fictional planets are written as proper nouns (e.g. Tatooine, Rigel IV, etc.), so I don't know how to strongly argue against proper-nounage in this case. It seems to be a name that refers to a very specific... something; it seems that those in this field would not ask "which Planet Nine do you mean?" so maybe it is not hard to argue that it rises to the level of a proper noun. dunno! $\endgroup$
    – uhoh
    Feb 10 at 3:12

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The planet would not be able to micro-lense light from distant stars, but it could in principle obscure a star along its path.

In order for gravitational lensing to work, the light path from the distant source has to be bent by a foreground object such that more light from the distant source arrives here. We call it microlensing, when a smaller body amplifies this effect by its own small gravitational field additional to another bigger source which it usually orbits. Thus the nearer the object is, the heavier it has to be in order to work as gravitational lens.

An object in our own solar system mostly interacts with distant sources via geometric optics, thus it obstructs the view. This is a occultation method or technique also used to determine the shape of asteroids (see e.g. here).

In our own solar system the gravitational lensing was used to confirm and test general relativity on our Sun during a total eclipse - but there are some recent measurements on Jupiter as well (e.g. Li et al 2022). That paper also lists needed measurement accuracies for the other planets, thus you can get an idea for the needed one for this alleged Planet IX.

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  • $\begingroup$ " the nearer the object is, the heavier it has to be in order to work as gravitational lens." Can't you just put the telescope further away from the object? The focal length of the Sun is 542 AU, don't smaller objects just have longer focal lengths, or is there a cut-off mass below which the light isn't bent at all? Not that this applies to detecting planet 9 from Earth, obviously. $\endgroup$ Feb 8 at 14:06
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    $\begingroup$ @DaveGremlin of course, the focal length "just" changes. But do you know how far already 542 AU are? Considering distances for telescopes for these purposes significantly different than 1AU from the Sun is science fiction. Neptun is 30 AU, the voyager probes are after 50 years of travel at about 160 AU, and the furthest known object in our solar system about 1000 AU, the next star is about 250.000 AU from us. $\endgroup$ Feb 8 at 14:14
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    $\begingroup$ The presence of a big planet would certainly lens a background star, in the sense of altering its apparent position. This is far more likely than an occultation. $\endgroup$
    – ProfRob
    Feb 8 at 15:57
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    $\begingroup$ @ProfRob that's exactly what I was thinking about. If we know there's an hypothetical set of orbital parameters, continous astrometric monitoring of background stars "in" the field of view of the expected orbit for this epoch could, in principle, say something about hypothetical microlensing, right? $\endgroup$
    – nuwe
    Feb 8 at 16:59
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With the caveat that it is easy to make an arithmetic blunder, I calculate that there is around a 1 in 25 chance per 6-month interval that planet 9 might cause a significant microlensing amplification of a star brighter than 21st magnitude. The problem is that nobody is really monitoring 21st-magnitude stars over a wide area on the sky, with a cadence sufficiently fast to catch the few-minute duration of such an event.

On the other hand, a measurable gravitational deflection is about 100 times more likely. i.e. Planet 9 is likely altering the positions of a few stars in its vicinity by a sufficient amount to be detectable by Gaia. By sifting through epoch-by-epoch positional measurements taken by the Gaia satellite we might be able to see its track across the sky in the form of a wave of perturbed stellar positions.

Details

(1) Microlensing Amplification

The size of any microlensing is characterised by the Einstein angle, given by $$\theta_E = \sqrt{\frac{4GM}{c^2}\left(\frac{d_s - d_l}{d_s d_l}\right)}\ , $$ where $M$ is the mass of the lensing object, $d_s$ is the distance to the background star and $d_l$ is the distance to the lensing object.

In this case $d_s - d_l \simeq d_s$ and so $$\theta_E = \sqrt{\frac{4GM}{c^2 d_l}}\ . $$

If we assume $M \sim 6 M_{\rm Earth}$ and $d_l \sim 500$ au, then $\theta_E \sim 0.008$ arcseconds. In order to get significant amplification then the background star and the planet need to be separated by an angle of this order or smaller.

There are about a billion catalogued stars down to around 21st magnitude, spread all over the 41,253 square degrees of sky, with positions measured to this kind of accuracy by Gaia, and with parallaxes and proper motions so that their sky positions can be predicted with this sort of precision into the (near) future. Thus the density of such stars is about 0.0019 per square arcsecond. At any one time, the chance of planet 9 being within its Einstein angle of a Gaia star is roughly $\pi (0.008)^2 \times 0.0019 \simeq 4 \times 10^{-7}$ (less than 1 in a million).

Of course, planet 9 will move. If we take a first-order approximation that it moves and the background stars are fixed, then it gets to roll the dice again every time it moves by 0.008 arcseconds on the sky. How fast does it move? At 500 au it will move at about 1 km/s, so to first order, its tangential motion is entirely due to the 30 km/s orbital motion of the Earth. In 6 months it will execute a parallax motion of about 800 arcseconds, so you have about $10^5$ chances for the Einstein angle to intercept a star and thus about a 1 chance in 25, over 6 months, that it will cause a microlensing event of some sort.

However, let me emphasize - this will be basically the chance that it is close to a star that is probably in the magnitude range 19-21, where the vast majority of these Gaia stars are. Nobody at the moment is monitoring all these stars to see if there is a microlensing event and since there is no well-determined likely location for planet 9, then a chance detection of this microlensing seems remote. The amplification event would only last a few minutes and so the cadence of any monitoring operation would have to be very high. It isn't going to be seen at the cadence of individual Gaia epochs (roughly 1 observation per month).

(2) Gravitational Deflection

Maybe a better bet is to look for systematic distortions of stellar positions for stars located close to planet 9 in the sky. These distortions are similar to those of stars being gravitationally lensed by the Sun. The deflection angle is approximately given by $$\theta_D \simeq \frac{4GM}{c^2 r}\ , $$ where $r$ is the distance of the closest approach of the light ray to the limb of planet 9.

Gaia is capable of measuring stellar positions to something like 1 milliarcsecond in a single visit for most stars it observes, and at least 10 times better for brighter stars. To get a deflection of this size, that could be compared with the average, and much more precisely determined, position of the star taken over the whole Gaia mission, would require $r \leq 2 \times 10^7$ m, corresponding to an angle on the sky at 500 au of 0.06 arcseconds and close to the limb of a Neptune-sized planet.

It is therefore about 100 times ($0.06^2/0.008^2$) more likely that a star will get close enough to planet 9 to cause gravitational deflection of measurable size than to cause significant microlensing amplification. Thus planet 9 would likely cause measurable deflections for several stars over the course of 6 months.

I would guess that it might be possible to sift through the epoch-by-epoch data from Gaia, looking for significant, transient, gravitational deflections and I would expect that several groups are probably gearing up to do that kind of analysis once the mission and its data processing nears completion.

(3) Occultation

Finally, there is the possibility that planet 9 will "eclipse" a background star, causing it to dim considerably. If its radius is of order $2 \times 10^7$ m (Neptune-sized). This is exactly the same size/angle as for the gravitational deflection calculation above, so the probability of it occuring is similar - i.e. maybe a few faint stars every 6 months. Again, the problem is, that these eclipses would last of order tens of minutes to 1 hour and nobody is monitoring the brightness of such faint stars over a wide area with that kind of cadence.

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    $\begingroup$ Tyvm for such a complete answer @ProfRob. "...would expect that several groups are probably gearing up to do that kind of analysis once the mission and its data processing nears completion". My BSc. thesis implied no observational work so I'm just being honestly ignorant about details on Gaia's data but, just for the sake of curiosity and fun, I'd like to approach this as a side project. Do you think that with the available data there are routines to work on? Maybe analysis can be done without the mission being completed. Could you, please, comment on hints to start throwing an "mvp" on this? $\endgroup$
    – nuwe
    Feb 9 at 0:42
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    $\begingroup$ @nuwe I'm not young enough to understand your acronyms. There is no publicly available data that would allow anyone to attempt this. e.g., no epoch-by-epoch Gaia data are available yet. $\endgroup$
    – ProfRob
    Feb 9 at 7:42
  • $\begingroup$ I apologize. Meant to say a "minimal viable product" (this is why I used "" because this is not exactly a product) but had no more space to write. Well. Although epoch-by-epoch data is not available, maybe the structure of Gaia's data is well known and the needed routines could be prepared. Do you know anything regarding to this? I'd be really pleased to tackle it in spare times $\endgroup$
    – nuwe
    Feb 9 at 16:27
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This was interesting so I did some calculations, not being an expert. A svelte rep is here

I used this formula to calculate the lensing effect for the sun as 1.746 and another page says the real value is 1.7520 prograde and 1.7519 retrograde so I'll call it good enough:

let angleSun = 4 * G * MSun / (c * c * rSun)

This article about the possible existance to explain a cluster of objects in the Kuiper belt with weird orbits mentions a Neptune-sized planet at 20 times the distance of Neptune, so I'll use that.

Calculating the lensing effect gives me a value of 0.00249 arc-seconds that it would deflect light. The actual visual size of planet 9 would be 0.11800 arc-seconds, so 0.05900 arc-seconds each direction from the center. That's about a factor of 24. So if it moved in front of a star we could theoretically see it eclipse the star, but it would eclipse the star and not magnify it. We would actually see the star for longer than normal because of the lensing effect as it passed behind the planet and that might lead us to think the planet was smaller than it is if we made the calculations without taking lensing into effect, but it wouldn't magnify, brighten, or create duplicates of objects behind it because it creates a blind spot.

The star with the largest angular diameter is https://en.wikipedia.org/wiki/R_Doradus with .051 arc-seconds, so even that would have no problem behind hidden by the planet.

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  • $\begingroup$ Just a thought: maybe someone should do (maybe in the future) a proposal to get this region monitored seeking for transient phenomena? or iterate over Gaia's data following what @ProfRob just added? $\endgroup$
    – nuwe
    Feb 8 at 23:35
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    $\begingroup$ That would be nice. We don't really have a region defined to look for it, probably somewhere near the ecliptic, but Pluto has over a 17 degree inclination, so if we take that as a limit that's a 35 degree band in all 360 degrees around the solar system. The Vera C. Rubin Observatory should find it if it's out there, just have to wait until 2026 to have enough observations :) $\endgroup$ Feb 9 at 5:35
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    $\begingroup$ In your code you have let G = 6.6743015e-11. That's wrong. Please see astronomy.stackexchange.com/a/48616/16685 $\endgroup$
    – PM 2Ring
    Feb 10 at 0:01
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    $\begingroup$ @PM2Ring Calling it 'wrong' is a little strong, but I get the point. I changed it to 6.6743. It's a technical mistake, but doesn't really change anything. According to your link there's a 68% chance it's between 6.67415 and 6.67445. $\endgroup$ Feb 11 at 23:08
  • $\begingroup$ Yes, 6.6743015 is within the range, so the results of calculating with it aren't wrong, per se. But my main concern was that you'd misread the notation, and that you believed we have a value of G with 8 significant figures, when we really only have 4 and a bit sig figs. As I said in the link, it's best to avoid calculating with G. $\endgroup$
    – PM 2Ring
    Feb 11 at 23:28

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