Planet classification

Hi,
here it comes, here it goes. Today I will write about planet classification since I want to start to write about dwarf planets. This post will be little similar to Stellar classification but you will see that for planets there are not those classes for size differences. The most important thing is of course to know what is planet and what is not, which I will explain.


Ok in the year of 2006 International Astronomical Union anounced that Pluto is not a planet. It`s been almost ten years but I still know people that can not get over it. I know this picture is sad (and not to scale).

For now the definition go as follows:

Planet has to be orbiting Sun
Have to be generally spherical
Has to have enough strong gravity to clear its orbit.

So the thing is that Pluto can only check first two criteria but not the third so such a object is called a dwarf planet.
Also you can see that planet is only object that is orbiting Sun which as it seems is not any exoplanet. This means that those 2000 planets that we found are just a huge objects, but for the official definition they are not planets either. And.. because they are so far away, it can not be known yet if they are spherical or not.

You can see that those definitions are not very good but luckily one guy on some conference proposed new definition which is not yet agreed to be new one but anyway International Astronomical Union will have to make a better one which will be probably very similar to this:

Planet has to be orbiting at least one star or the remains of one. (So yeah, stuff that is orbiting white dwarf is still a planet if it meets the other criteria. Read about multiple star systems)
Planet has to have a clear path to itself. (Sorry Pluto.)
Planet has to have mass lower than the mass of Jupiter. (This is good so we dont accidentaly name brown dwarfs as planets.)

Now we know what are planets. Lets move what are dwarf planets. Those are the objects that are not able to satisfy the third rule, their orbit around Sun is not clean. Dwarf planets are: CeresCeres, the only dwarf planet in the asteroid belt imaged by Dawn, PlutoPluto seen by New Horizons on 13 July 2015, HaumeaHaumea with its two moons, as seen by Keck, MakemakeMakemake imaged by the Hubble Telescope in 2006, ErisEris and its moon seen from Hubble and Sedna for example.Sedna seen through HubbleThere are some other candidates also.

There is type of objects that are called minor planets. Those can be at the same time dwarf planets, like Ceres. The number of minor planets is increasing by very large amount every month. Only few of them are named, large part is numbered and there is  rest for which we dont even know exact orbit. Together there are almost 700,000 minor planets.

I did not find exact definition but minor planets should be those that are orbiting Sun, that is about it. They dont have to have any particular size, shape and their path can be trafic jam of asteroids. Actually yes, the asteroids are minor planets and also all the trojans and so on.
Above you can see all the types of objects that can appear in our Solar System except Sun.
Thats about it, I will definitely make post about or more about asteroids, comets and of course I will be continuing with moons.

Dragallur

Pictures are from wikipedia pages: IAU definition of planets, Minor planets and Dwarf planets and the first one is from this page:

Turn your brain into supermassive black hole

Hi,
here it comes here it goes! Today I am writing probably the last post from Brazil since I am leaving in Tuesday. Maybe tommorow after seeing Christ in Rio, I will be able to write another one.
Anyway, here comes the promised post about Ackermann function.

Ackermann function is a function with two inputs and it is growing extremely fast. To see the results it is best to make a table. Ackermann
function is written like this: A(m,n).
There are few types of Ackermann function because some people made their modifications of it to fit their plans. Here I will mention the most famous one.
 A(m, n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0. \end{cases} So lets see what this function means:
A(m,n) is the input of this. If m=0 then you will make n+1 and find it in the table which is below.
If m is bigger than 0 and n=0 then you will call Ackermann function again with m-1 and n=1.
Last one, when you have m>0 and n>0 then you call the function again with m lowered by one and n will be defined by another A function which obeys

m\n 0 1 2 3 4 n
0 1 2 3 4 5 n + 1

the rules again. Lets see the table. This is the first row which is extremely simple. M is the vertical axis. So when the arguments are A(0,0) then you go as follows: m=0 which means that you higher n by one which is 1 and that is the result.

If A(3,2) it gets very messy: here you go.
A(3,2)
A(2,A(3,1)  — because n and m are higher than 0 you lower m by one and then you call another function with n lowed by one.
A(2,A(2,A(3,0) — both are still higher, so you do the same thing again..
A(2,A(2,A(2,1) — change the function from A(3,0) to A(2,1) because n is 0 according to rule above
A(2,A(2,A(1,A(2,0)
A(2,A(2,A(1,A(1,1)
A(2,A(2,A(1,A(0,A(1,0)
A(2,A(2,A(1,A(0,A(0,1)
A(2,A(2,A(1,A(0,2)
A(2,A(2,A(1,3)
A(2,A(2,A(0,A(1,2)
A(2,A(2,A(0,A(0,A(1,1)
A(2,A(2,A(0,A(0,A(0,A(1,0)
A(2,A(2,A(0,A(0,A(0,A(0,1)
A(2,A(2,A(0,A(0,A(0,2)
A(2,A(2,A(0,A(0,3)
A(2,A(2,A(0,4)
A(2,13) — I skipped lot of the steps because it is such a mess when you have to do it whole again but it equals 29.
=29

Values of A(mn)
m\n 0 1 2 3 4 n
0 1 2 3 4 5 n + 1
1 2 3 4 5 6 n + 2 = 2 + (n + 3) - 3
2 3 5 7 9 11 2n + 3 = 2\cdot(n + 3) - 3
3 5 13 29 61 125 2^{(n + 3)} - 3
4 13

={2^{2^{2}}}-3

65533

={2^{2^{2^{2}}}}-3

265536 − 3

={2^{2^{2^{2^{2}}}}}-3

{2^{2^{65536}}} - 3

={2^{2^{2^{2^{2^{2}}}}}}-3

{2^{2^{2^{65536}}}} - 3

={2^{2^{2^{2^{2^{2^{2}}}}}}}-3

\begin{matrix}\underbrace{{2^2}^{{\cdot}^{{\cdot}^{{\cdot}^2}}}} - 3\\n+3\end{matrix}
5 65533

=2\uparrow\uparrow\uparrow 3 - 3

2\uparrow\uparrow\uparrow 4 - 3 2\uparrow\uparrow\uparrow 5 - 3 2\uparrow\uparrow\uparrow 6 - 3 2\uparrow\uparrow\uparrow 7 - 3 2\uparrow\uparrow\uparrow (n+3) - 3
6 2\uparrow\uparrow\uparrow\uparrow 3 - 3 2\uparrow\uparrow\uparrow\uparrow 4 - 3 2\uparrow\uparrow\uparrow\uparrow 5 - 3 2\uparrow\uparrow\uparrow\uparrow 6 - 3 2\uparrow\uparrow\uparrow\uparrow 7 - 3 2\uparrow\uparrow\uparrow\uparrow (n+3) - 3

Here is the rest of the start of the table. It increases rapidly since it repeats over and over again (this is called recursion). You see that you have to use Knuth`s up-arrow notation.
Now you see why A(G64,G64) is such spawn of hell.
The reason for Ackermann function to exist and to be so famous is that it is one of the first functions that are used in computability theory. It is the theory which asks what means that the function is not computable and how much not computable it is. Computable functions are those for which we can find some algorithms, and algorithms are very important. For example in computations.
It seems for example that there is no computable function for finding prime numbers or at least no efficient one.

Dragallur