March 4, 2026
only from m9 = 5 to m9 = 6 M⊕
and from a9 = 300 to a9 =
310 AU. The orbital angles do not change substantially.
We conclude that the preference for smaller values of mass and
semimajor axis is robust, and that the orbital angles (i9,
Ω9, $9) are largely
unaffected by any contamination. While the posterior distributions
for m9 and a9 have large tails
towards larger values, the possibility of a closer brighter Planet
Nine needs to be seriously considered.
An additional uncertainty worth considering is the diameter and
albedo of Planet Nine. We have assumed values appropriate for a
gas-rich sub-Neptune which, a priori, seems the most likely
state for such a distant body. Given our overall ignorance of the
range of possibilities in the outer solar system, we cannot exclude
the possibility of an icy body resembling, for ex- ample, a
super-Eris. Such an icy/rocky body
could be ∼50%
smaller than an equivalent sub- Neptune in this mass range (Lopez
&
Fortney
2014), and
while the large KBOs like Eris have high albedos, much of this
elevated albedo could be driven by frost covering of darker
irradiated materials as the objects move through very dif- ferent
temperature regimes on very eccentric or- bits. An object at the
distance of Planet Nine – which stays below the condensation
tempera- ture of most volatiles at all times – could well lack such
volatile recycling and could have an albedo closer to the ∼10%
of the large but not volatile-covered KBOs (Brown
2008).
Overall the effect of a smaller diameter and smaller albedo could
make Planet Nine ∼ 3
magni- tudes dimmer. Such a situation would make the search for
Planet Nine considerably more difficult. While the possibility of a
dark super- Eris Planet Nine seems unlikely, it cannot be excluded.
Finally, we recall the affect of the choice of the prior on a9.
A prior assuming formation in a cluster would put Planet Nine more
distant
than shown here, though it would also predict higher masses.
Combining those effects we find that the magnitude distribution seen
in Figure 8 would shift fainter by about a magnitude near aphelion
but would change little close to peri- helion.
While all of these caveats affect the distance, mass, and brightness
of Planet Nine, they have no affect on the sky plane position shown
in Figure 8. To a high level of confidence, Planet Nine should be
found along this delineated path.
CONCLUSION
We have presented the first estimate of Planet Nine’s mass and
orbital elements using a full statistical treatment of the likelihood
of detec- tion of the 11 objects with 150 < a
< 1000 AU and q > 42 AU as well
as the observa- tional biases associated with these detections. We
find that the median expected Planet Nine semimajor axis is
significantly closer than previ- ously understood, though the range
of potential distances remains large. At its brightest pre- dicted
magnitude, Planet Nine could well be in range of the large number of
sky surveys being performed with modest telescope, so we expect that
the current lack of detection suggests that it is not as the
brightest end of the distribution, though few detailed analysis of
these surveys has yet been published.
Much of the predicted magnitude range of Planet Nine is within the
single-image detec- tion limit of the LSST survey of the Vera Rubin
telescope, r ∼ 24.3,
though the current survey plan does not extend as far north as the
full pre- dicted path of Planet Nine. On the faint end of the
distribution, or if Planet Nine is unexpect- edly small and dark,
detection will still require
imaging with 10-m class telescopes or larger.
Despite recent discussions, statistical evidence for clustering in
the outer solar system remains strong, and a massive planet on a
distant in- clined eccentric orbit remains the simplest hy- pothesis.
Detection of Planet Nine will usher in a new understanding of the outermost part of our solar system and
allow detailed study of a fifth giant planet with mass common
through- out the galaxy.
ACKNOWLEDGMENTS
This manuscript owes a substantial debt to the participants at the
MATH + X Sympo- sium on Inverse Problems and Deep
Learning in Space Exploration held at Rice
University in Jan 2019 with whom we discussed the issue of inverting
the observations of KBOs to solve for Planet Nine. We would also like
to thank two anonymous reviewers of a previous paper whose excellent
suggestions ended up being incorpo- rated into this paper and @Snippy
X
and @si-
welwerd
on Twitter for advice on notation for our likelihood
functions.
Software: HEALPix
(Gorski et al.
2005), as- tropy (Astropy
Collaboration
et
al.
2013),
scikit- learn (Pedregosa
et
al.
2011),
emcee (Foreman-
Mackey et al.
2013), corner (Foreman-Mackey
2016)
Table
2.
-
|
m9
(Mearth)
|
a9
(AU)
|
i9
(deg)
|
e9
|
a9
(deg)
|
Ω9
(deg)
|
l
|
∆l
|
num.
particles
|
|
3
|
625
|
15
|
0.60
|
356
|
166
|
-182.1
|
-9.2
|
21100
|
|
4
|
230
|
10
|
0.15
|
250
|
108
|
-175.5
|
-2.6
|
30000
|
|
4
|
250
|
15
|
0.15
|
260
|
102
|
-175.3
|
-2.4
|
30000
|
|
4
|
500
|
20
|
0.33
|
224
|
86
|
-176.2
|
-3.3
|
120500
|
|
5
|
230
|
10
|
0.15
|
246
|
96
|
-174.3
|
-1.4
|
30000
|
|
5
|
250
|
5
|
0.15
|
250
|
126
|
-177.0
|
-4.1
|
30000
|
|
5
|
250
|
10
|
0.15
|
248
|
108
|
-174.4
|
-1.5
|
30000
|
|
5
|
260
|
15
|
0.10
|
246
|
94
|
-174.2
|
-1.3
|
25600
|
|
5
|
260
|
5
|
0.15
|
246
|
82
|
-177.0
|
-4.1
|
30000
|
|
5
|
280
|
10
|
0.10
|
246
|
96
|
-175.8
|
-2.9
|
25600
|
|
5
|
280
|
15
|
0.10
|
266
|
88
|
-175.0
|
-2.1
|
25600
|
|
5
|
300
|
10
|
0.15
|
234
|
108
|
-175.6
|
-2.7
|
25600
|
|
5
|
300
|
17
|
0.15
|
254
|
108
|
-172.9
|
0.0
|
25600
|
|
5
|
310
|
15
|
0.10
|
274
|
102
|
-175.1
|
-2.2
|
25600
|
|
5
|
356
|
17
|
0.20
|
252
|
88
|
-174.2
|
-1.3
|
25600
|
|
5
|
500
|
5
|
0.33
|
250
|
96
|
-179.2
|
-6.3
|
25600
|
|
5
|
500
|
10
|
0.33
|
244
|
86
|
-176.1
|
-3.2
|
25500
|
|
5
|
500
|
20
|
0.33
|
234
|
86
|
-176.2
|
-3.3
|
20200
|
|
5
|
720
|
20
|
0.65
|
234
|
96
|
-185.1
|
-12.2
|
30100
|
|
6
|
280
|
17
|
0.10
|
256
|
100
|
-173.2
|
-0.3
|
25500
|
|
6
|
290
|
17
|
0.15
|
250
|
108
|
-173.0
|
-0.0
|
25600
|
|
6
|
300
|
17
|
0.15
|
246
|
100
|
-173.4
|
-0.4
|
25600
|
|
6
|
310
|
10
|
0.10
|
252
|
96
|
-174.4
|
-1.5
|
25600
|
|
6
|
310
|
15
|
0.10
|
256
|
96
|
-174.6
|
-1.7
|
25600
|
|
6
|
310
|
17
|
0.10
|
244
|
108
|
-175.0
|
-2.1
|
25600
|
|
6
|
310
|
10
|
0.15
|
256
|
108
|
-173.0
|
-0.1
|
25600
|
|
6
|
310
|
15
|
0.15
|
252
|
116
|
-173.0
|
-0.1
|
25600
|
|
6
|
310
|
17
|
0.15
|
266
|
106
|
-173.5
|
-0.6
|
19900
|
|
6
|
310
|
5
|
0.20
|
244
|
108
|
-177.1
|
-4.2
|
25600
|
|
6
|
310
|
10
|
0.20
|
244
|
108
|
-173.9
|
-1.0
|
25000
|
|
6
|
310
|
15
|
0.20
|
252
|
92
|
-173.0
|
-0.0
|
25400
|
Table
2
continued
-
|
m9
(Mearth)
|
a9
(AU)
|
i9
(deg)
|
e9
|
a9
(deg)
|
Ω9
(deg)
|
l
|
∆l
|
num.
particles
|
|
6
|
310
|
17
|
0.20
|
260
|
122
|
-173.2
|
-0.3
|
13600
|
|
6
|
310
|
20
|
0.20
|
242
|
96
|
-173.2
|
-0.3
|
23700
|
|
6
|
310
|
25
|
0.20
|
230
|
92
|
-174.7
|
-1.8
|
20000
|
|
6
|
310
|
30
|
0.20
|
238
|
88
|
-178.0
|
-5.1
|
25500
|
|
6
|
330
|
10
|
0.20
|
248
|
108
|
-174.6
|
-1.7
|
31300
|
|
6
|
330
|
15
|
0.20
|
252
|
92
|
-173.4
|
-0.5
|
14400
|
|
6
|
356
|
20
|
0.10
|
254
|
100
|
-175.3
|
-2.4
|
25600
|
|
6
|
356
|
20
|
0.15
|
250
|
110
|
-174.2
|
-1.3
|
25600
|
|
6
|
356
|
15
|
0.20
|
256
|
102
|
-174.1
|
-1.2
|
21200
|
|
6
|
356
|
17
|
0.20
|
262
|
100
|
-174.1
|
-1.2
|
25600
|
|
6
|
356
|
17
|
0.20
|
264
|
108
|
-173.9
|
-1.0
|
25600
|
|
6
|
356
|
19
|
0.20
|
238
|
100
|
-173.9
|
-1.0
|
48500
|
|
6
|
356
|
25
|
0.20
|
228
|
88
|
-176.2
|
-3.3
|
40200
|
|
6
|
356
|
30
|
0.20
|
238
|
96
|
-179.9
|
-6.9
|
16700
|
|
6
|
380
|
17
|
0.20
|
242
|
110
|
-174.1
|
-1.2
|
25600
|
|
6
|
380
|
17
|
0.25
|
246
|
92
|
-173.3
|
-0.3
|
25600
|
|
6
|
500
|
35
|
0.15
|
242
|
96
|
-181.8
|
-8.9
|
30000
|
|
6
|
600
|
40
|
0.15
|
260
|
94
|
-184.0
|
-11.1
|
30000
|
|
6
|
800
|
50
|
0.15
|
242
|
82
|
-188.4
|
-15.5
|
30000
|
|
7
|
356
|
17
|
0.20
|
246
|
92
|
-173.8
|
-0.9
|
25600
|
|
7
|
400
|
15
|
0.25
|
254
|
82
|
-173.9
|
-1.0
|
30900
|
|
7
|
400
|
20
|
0.25
|
246
|
102
|
-175.2
|
-2.3
|
52800
|
|
7
|
400
|
30
|
0.25
|
230
|
88
|
-177.5
|
-4.6
|
30800
|
|
7
|
450
|
25
|
0.15
|
248
|
108
|
-178.7
|
-5.8
|
30000
|
|
7
|
450
|
15
|
0.33
|
250
|
86
|
-175.8
|
-2.8
|
29700
|
|
7
|
450
|
20
|
0.33
|
236
|
80
|
-175.9
|
-3.0
|
25600
|
|
7
|
450
|
25
|
0.33
|
236
|
80
|
-176.2
|
-3.3
|
23500
|
|
7
|
500
|
20
|
0.15
|
256
|
94
|
-176.3
|
-3.4
|
25600
|
|
7
|
500
|
15
|
0.20
|
256
|
102
|
-175.6
|
-2.7
|
25600
|
|
7
|
500
|
17
|
0.20
|
268
|
96
|
-175.1
|
-2.1
|
25600
|
|
7
|
500
|
25
|
0.20
|
254
|
92
|
-177.6
|
-4.7
|
25600
|
|
7
|
500
|
20
|
0.25
|
260
|
94
|
-176.8
|
-3.9
|
25600
|
|
7
|
500
|
5
|
0.33
|
242
|
96
|
-178.2
|
-5.2
|
57300
|
Table
2
continued
Table
2
(continued)
-
|
m9
(Mearth)
|
a9
(AU)
|
i9
(deg)
|
e9
|
a9
(deg)
|
Ω9
(deg)
|
l
|
∆l
|
num.
particles
|
|
7
|
500
|
10
|
0.33
|
252
|
92
|
-176.6
|
-3.7
|
41400
|
|
7
|
500
|
15
|
0.33
|
250
|
98
|
-175.5
|
-2.6
|
47700
|
|
7
|
500
|
17
|
0.33
|
250
|
100
|
-175.4
|
-2.5
|
17500
|
|
7
|
500
|
20
|
0.33
|
242
|
86
|
-176.1
|
-3.2
|
52400
|
|
7
|
500
|
25
|
0.33
|
234
|
86
|
-177.9
|
-5.0
|
54000
|
|
7
|
500
|
30
|
0.33
|
232
|
94
|
-179.0
|
-6.1
|
59600
|
|
7
|
500
|
35
|
0.33
|
230
|
86
|
-180.5
|
-7.6
|
41700
|
|
7
|
500
|
25
|
0.40
|
228
|
86
|
-179.7
|
-6.8
|
35000
|
|
7
|
500
|
25
|
0.45
|
226
|
74
|
-182.0
|
-9.0
|
27700
|
|
7
|
525
|
20
|
0.50
|
236
|
70
|
-179.6
|
-6.6
|
33000
|
|
7
|
550
|
17
|
0.40
|
244
|
88
|
-175.6
|
-2.6
|
25600
|
|
7
|
600
|
17
|
0.45
|
238
|
94
|
-174.9
|
-2.0
|
25600
|
|
7
|
640
|
17
|
0.50
|
240
|
102
|
-176.8
|
-3.9
|
16900
|
|
7
|
650
|
17
|
0.45
|
230
|
88
|
-174.6
|
-1.7
|
25500
|
|
7
|
800
|
50
|
0.15
|
310
|
50
|
-190.4
|
-17.5
|
30000
|
|
7
|
830
|
20
|
0.70
|
208
|
96
|
-184.7
|
-11.7
|
51200
|
|
7
|
1000
|
60
|
0.15
|
298
|
94
|
-191.2
|
-18.3
|
30000
|
|
8
|
400
|
20
|
0.15
|
248
|
108
|
-177.1
|
-4.2
|
30000
|
|
10
|
350
|
10
|
0.15
|
250
|
96
|
-176.3
|
-3.4
|
30000
|
|
10
|
400
|
20
|
0.15
|
242
|
84
|
-178.2
|
-5.3
|
30000
|
|
10
|
450
|
20
|
0.33
|
242
|
82
|
-177.8
|
-4.9
|
34300
|
|
10
|
525
|
20
|
0.15
|
264
|
106
|
-178.1
|
-5.2
|
30000
|
|
10
|
525
|
30
|
0.15
|
266
|
102
|
-184.6
|
-11.7
|
30000
|
|
10
|
525
|
40
|
0.15
|
304
|
138
|
-189.9
|
-17.0
|
30000
|
|
10
|
525
|
20
|
0.50
|
244
|
114
|
-180.8
|
-7.9
|
39700
|
|
10
|
525
|
20
|
0.65
|
242
|
90
|
-181.7
|
-8.8
|
20900
|
|
10
|
525
|
30
|
0.65
|
244
|
36
|
-187.1
|
-14.2
|
35600
|
|
10
|
700
|
20
|
0.35
|
244
|
108
|
-176.6
|
-3.7
|
25600
|
|
10
|
700
|
30
|
0.70
|
290
|
132
|
-190.0
|
-17.1
|
25600
|
|
10
|
750
|
10
|
0.35
|
234
|
106
|
-177.5
|
-4.6
|
19500
|
|
10
|
750
|
15
|
0.35
|
252
|
114
|
-176.1
|
-3.2
|
22400
|
|
10
|
750
|
20
|
0.35
|
244
|
100
|
-177.9
|
-5.0
|
25500
|
|
10
|
800
|
5
|
0.40
|
244
|
114
|
-177.5
|
-4.6
|
25600
|
Table
2
continued
-
|
m9
(Mearth)
|
a9
(AU)
|
i9
(deg)
|
e9
|
a9
(deg)
|
Ω9
(deg)
|
l
|
∆l
|
num.
particles
|
|
10
|
800
|
10
|
0.40
|
240
|
112
|
-177.0
|
-4.1
|
25600
|
|
10
|
800
|
15
|
0.40
|
240
|
118
|
-177.8
|
-4.9
|
25600
|
|
10
|
800
|
15
|
0.45
|
240
|
120
|
-174.9
|
-2.0
|
25600
|
|
10
|
800
|
20
|
0.45
|
238
|
108
|
-176.0
|
-3.1
|
28600
|
|
10
|
800
|
25
|
0.45
|
234
|
100
|
-177.6
|
-4.7
|
23500
|
|
10
|
800
|
30
|
0.45
|
242
|
50
|
-184.0
|
-11.1
|
16800
|
|
10
|
800
|
60
|
0.45
|
182
|
114
|
-183.0
|
-10.1
|
30400
|
|
10
|
870
|
20
|
0.73
|
254
|
92
|
-185.4
|
-12.5
|
17900
|
|
10
|
1000
|
60
|
0.15
|
314
|
96
|
-192.8
|
-19.9
|
23600
|
|
10
|
1400
|
70
|
0.15
|
224
|
30
|
-190.0
|
-17.1
|
30000
|
|
12
|
500
|
15
|
0.20
|
256
|
94
|
-178.4
|
-5.5
|
25600
|
|
12
|
500
|
20
|
0.20
|
256
|
92
|
-181.2
|
-8.3
|
25600
|
|
12
|
500
|
25
|
0.20
|
266
|
102
|
-182.9
|
-10.0
|
25600
|
|
12
|
920
|
20
|
0.73
|
224
|
76
|
-182.1
|
-9.1
|
25800
|
|
12
|
960
|
20
|
0.79
|
242
|
54
|
-186.8
|
-13.9
|
24900
|
|
14
|
960
|
20
|
0.74
|
220
|
76
|
-185.6
|
-12.7
|
28000
|
|
16
|
1000
|
20
|
0.75
|
248
|
76
|
-183.2
|
-10.2
|
33600
|
|
20
|
900
|
60
|
0.15
|
306
|
66
|
-189.0
|
-16.1
|
30000
|
|
20
|
1000
|
15
|
0.65
|
242
|
122
|
-179.6
|
-6.7
|
30100
|
|
20
|
1000
|
20
|
0.65
|
240
|
118
|
-180.6
|
-7.7
|
33000
|
|
20
|
1000
|
25
|
0.65
|
246
|
70
|
-185.5
|
-12.6
|
32300
|
|
20
|
1070
|
20
|
0.77
|
240
|
124
|
-185.2
|
-12.3
|
64900
|
|
20
|
1400
|
70
|
0.15
|
264
|
0
|
-186.8
|
-13.9
|
30000
|
|
20
|
2000
|
80
|
0.15
|
260
|
152
|
-190.1
|
-17.2
|
30000
|
Note—Parameters
used in the numerical simulations on the effects of Planet Nine (m9,
a9,
i9,
e9)
and the maximum ln(likelihood), l,
which occurs at the listed value
of
a9
and
Ω9.
∆l
gives
the
difference
in
ln(likelihood)
from
the
maximum value, which
occurs at m9
=
5, a9
=
310, i9
=
15, and e9
=
0.10.
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