... raise questions about how much we know about the true (IAU definition) number of planets in our solar system.
Eight.
Question: Does the discovery of 2012 VP$_{113}$ and 90377 . increase the odds of some kind of planetary object on the edge of the known solar system (the Oort cloud)?
2012 VP$_{113}$ is a minor planet, that does not imply that it is a "planet".
90377 Sedna is a large minor minor planet, also not a "planet".
The Oort cloud has so many tiny objects that they upset each other's orbit, occasionally sending comets into our solar system. Part of being a planet is having an orbit and clearing your orbit; these objects do neither.
How could we find out if it is very cold and dark?
Every object in space affects the movement of every other object.
We can detect tiny black holes at a distance, they are cold and dark; detecting a new planet in our solar system isn't beyond our ability, there simply isn't one there.
It's not impossible for a large object to begin orbit of the sun and establish that it will orbit a reasonably fixed path (some wobble is OK, drastic changes in direction or too close to something large enough isn't going to maintain it's orbit (and clearing ability)).
Pluto takes 248.00 years (90,560 days) to complete an orbit. Discovered in 1930 it lost its status as a planet in 2006. If something appears out of nowhere it will take time for it to be approved, so no surprises in the next few years.
I mean, how many 90377 Sedna or 2012 VP$_{113}$ objects would we need in order to guess its position and location?
Objects very distant to each other affect each other to a much smaller extent that objects which are closer to each other. As time passes, observations and new calculations are made, results are confirmed, and it passes IAU committee approval something might be discovered and become a new planet, instead scientists are looking outside our solar system for exoplanets.
Beta Pictoris (β Pic) is the second brightest star in the constellation Pictor. It is located 63.4 light years from the Solar System. Scientists believe there are a few planets in orbit around the star.
"The planet companion around β Pictoris" (Oct 2010), by Anne-Marie Lagrange, Mickaël Bonnefoy, Gael Chauvin, Daniel Apai, Et al.:
Abstract
The β Pic disk of dust and gas has been regarded as the prototype of young planetary systems since the 1980s and has revealed over the years an impressive amount of indirect signs pointing toward the presence of at least one giant planet. We present here the recently detected first giant planet around this star. We show how this planet could explain some very peculiar features of the star environment (disk, spectroscopic variability), and how it constrains the scenarios of planetary system formation (timescales, mechanisms).
Scientists can see far and do look for planets, it's unlikely that there's a 11th planet that's been unobserved. They have nicknamed β Pic's planet "b". Here's its photo:
Reading about this topic, I don't know how to quantify the odds of finding:
There is support for the existence of the object, but under the IAU definition imposed by your question it hasn't yet met the requirements to become a planet. The announcement does not mean there is a new planet in our solar system. The existence of this distant world is only theoretical at this point and no direct observation of the object nicknamed have been made. The mathematical prediction of a planet could explain the unique orbits of some smaller objects in the Kuiper Belt, a distant region of icy debris that extends far beyond the orbit of Neptune. Astronomers are now searching for the predicted planet.
- Thelistos: A super-Earth (about 5-10 Earth masses times a O(1) factor)
Misnomer. Deleted from Wikipedia, redirects to planet X.
- Tyche: (Now disfavoured by WISE data, I think it can not be excluded if it is extremely cold and "dark") Gas giant (Neptune-like, Saturn-like, etc.)
Disproven. Does not exist.
- Nemesis: A "dark star"/brown dwarf object.
128 Nemesis is an asteroid, the other Nemesis is a hypothetical red dwarf or brown dwarf, originally postulated in 1984 to be orbiting the Sun at a distance of about 95,000 AU (1.5 light-years). Thought to be somewhat beyond the Oort cloud, it's postulated to explain a perceived cycle of mass extinctions in the geological record, which seem to occur more often at intervals of 26 million years.
Which objects are defined as planets is precisely defined, there's only some overlap for non-planets:
Here is the text of the IAU’s Resolution B5: Definition of a Planet in the Solar System (.PDF):
The IAU therefore resolves that planets and other bodies, except satellites, in our Solar System be defined into three distinct categories in the following way:
(1) A planet$^1$ is a celestial body that
$\quad$ (a) is in orbit around the Sun,
$\quad$ (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
$\quad$ (c) has cleared the neighbourhood around its orbit.
(2) A "dwarf planet" is a celestial body that
$\quad$ (a) is in orbit around the Sun,
$\quad$ (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape$^2$,
$\quad$ (c) has not cleared the neighbourhood around its orbit, and
$\quad$ (d) is not a satellite.
(3) All other objects$^3$, except satellites, orbiting the Sun shall be referred to collectively as "Small Solar System Bodies".
1 The eight planets are: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune.
2 An IAU process will be established to assign borderline objects to the dwarf planet or to another category.
3 These currently include most of the Solar System asteroids, most Trans-Neptunian Objects (TNOs), comets, and other small bodies.
