# Thread: size of a black hole

1. I was trying to figure out the necessary weight and size an object has to be to become a black hole and this is what I did. But I have a small unit related problem that is blatantly obvious. I know where the problem is but I don't know how to solve it.

F = (Gmamb) / r^2

where F = force
G = the gravitational constant... 6.67E-11 m^3/Kg*s^2
ma = mass of object a in kilograms
mb = mass of object b in kilograms
and r = the distance between object a's and object b's centers of gravities in meters

You can figure out the acceleration an object will cause ANY another object of any weight to experience...

a = acceleration

a = f/mb = (Gmamb) / mbr^2 = (Gma)/r^2

I then plugged the speed of light in for a because the black hole's gravity has to pull light in faster than the speed of light in order to be a black hole.

300 000 = 6.67E-11m/r^2
2.68E+8 = m/r^2

m is in kilograms
r is in meters

So if an object is more than 2.68E+8Kg/m^2 then the object is a black hole.

Too me it seems that it should be 2.68Kg/m^3 instead... I know the problem lies with my putting in the speed of light for acceleration because the speed of light is a velocity, so that causes me to lose a second unit on the left side of my equation but I also think there is something else wrong with my units... But I am fairly sure my number is right. How should I go about solving this? Is there something else I should put in instead of the speed of light? Thanks.

2.

3. Hi Shawngoldw

Ignoring the problem of using Newtonian physics to talk about blackholes (the actual treatment is a lot more technical and uses GR) you are going about this the wrong way entirely.

What is your maths and physics background? I'm asking this to find out if you know about gravitational fields and energy which is the "correct" way to derive the escape velocity equation for a body of mass M and radius r.

4. o... well my math and physics background is pretty weak... I just finished grade 10 with a 92 in science and an 86 in gifted math... I have spent a lot of time though trying to learn this stuff so I know a bit about relativity, mostly special though not general. I'll look into checking this out using GR though.

5. Ok, so is that good or bad and do you know about gravitational fields and conservation of mechanical energy?

6. i know a bit about gravitation fields but i dont know about conservation of mechanical energy. is it related to conservation of mass and energy? because I know about that.

7. There are two ways to get the escape velocity of a body of mass M and radius R<sub>0</sub>. The first way is to use conservation of energy, which you should be able to do given your background.

The other way is to solve the ODE r'' = GM/r<sup>2</sup>

8. thanks. I used that formula in my first post. but i used a instead of r". and whats the ODE?

9. Yes but you used it incorrectly. do you know what an ODE is?

10. nope

11. Originally Posted by shawngoldw
I was trying to figure out the necessary weight and size an object has to be to become a black hole and this is what I did.
Scientists have beat you to it, though I can imagine how fun it might be to try. Aside from that, you mean figure out the minimal mass required for a star to become a black hole. This is known as the "Chandrasekhar Limit" (unless I missed my research).

http://en.wikipedia.org/wiki/Chandrasekhar_limit

Some information on it can be found there, but due to my lack of professional education I can't offer much other than wiki articles (I advise you read them until they make sense).

http://en.wikipedia.org/wiki/Black_hole

12. I wasn't sure what they called it so thanks for that piece of info. but thats exactly it, i wanna try to figure it out. And does anyone want to tell me what an ode is?

13. http://en.wikipedia.org/wiki/ODE

learn to use wikipedia and google. The one you want is: "Ordinary differential equation, a mathematical concept."

14. i used google and was getting things like... open dynamics engine, some magazine, ohio department of education and a writers guide. I had no idea what I was looking for. Thanks for the link.

15. Alright, this is what I got... The ODE is
f(x) = m(dv/dt).

But that is for Newtonian physics. When you take special relativity into account you get
F = yma

m is the rest mass and y is supposed to be a symbol but I don't know how to type it and I don't know what its called. It looks almost like a y.
its equation is...
y = 1/(1 - v^2/c^2)

So I use this to get force instead of F = m/a but to get acceleration do I still use a = Gm/r^2 ? Because I found this on wikipedia
G is the gravitation constant, r is distance and r with a arrow thing on it is a unit vector identifying the direction to the massive object. But it doesn't say what the r with the dots is. The formula looks very much like my old formula. Are they the same?

16. Math aside, an object is going to become a black hole only if its mass is sufficiently large to overcome the outward forces keeping the mass from collapsing. This includes the atomic electromagnetic forces that keep atoms separated. Once this is overcome, there's nothing stopping the object from collapsing upon itself.

If the mass is too small, the object will not become a black hole. Stars below 1.5 solar mass will implode, but explode their mass into space afterwards, rather than creating a black hole. Stars with a larger mass may become black holes.

They can also become neutron stars, if the mass is large enough to overcome the atomic outward forces, but not the forces of the neutrons within the atoms.

The state of the object has some bearing as well. In the highly-unlikely scenario of a massive object that isn't moving or rotating, it is possible that collapse could occur sooner than in a mass that is spinning.

17. Don't worry about the ODE method Shaungoldw, i'll step you through the conservation of energy approach.

Now the potential energy of a point mass m in a gravitational field GM/r<sup>3</sup>r is U = -GMm/r where r is the vector from the source of the field to the point particle in question and r is the length of that vector. This is just newtons law of universal gravitation all dressed up in fancy clothes btw.

Now the total mechanical energy (i.e. the sum of the kinetic and potential energy of a particle) is conserved so we have that:

1/2 m v<sup>2</sup> - GMm/r = Constant

The total energy of the particle must remain positive, which limits how far away the particle can move from the source of the field. Now we want our particle to have positive energy no matter how far away from the source which is the same as saying that it can escape the field and "reach infinity". So our constant must be greater or equal then zero, so solving for v<sup>2</sup> we get that

v<sup>2</sup> >= 2GM/R<sub>0</sub>

Where R<sub>0</sub> is the starting distance of the particle from the source (which would be the radius of the ball of matter in our case). Now lets suppose that we cannot exceed the speed of light, so our maximum escape velocity v = c. Substituting into the above equation we arrive at the Schwarzschild radius

R<sub>s</sub> = 2GM/c<sup>2/sup>

Which would be the distance from the singularity of the event horizon of a non-rotating uncharged black hole (if this derivation made any real sense in that situation, which it doesn't)

Its strange that the maths works and the more complicated GR method gives the same result, but the physics is entirely wrong!

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