Estimate the radius of a solar mass black hole.

[hasan_researc]

Estimate the radius of a solar mass (2 × 1030 kg) black hole.

How do I do it? I have literally no idea!

Thanks in advance for any help.

 

[Danger]

That's a trick question. A star has to be of at least 3.2 solar masses in order to collapse into a black hole. There might theoretically be microscopic "primordial" black holes, but they wouldn't mass anything like a star. In any event, the radius of a black hole, regardless of mass, is zero. That, of course, refers to the singularity which is the actual heart of the hole.
There are other radii which pertain to a black hole. One is the "event horizon". That is the point at which escape speed exceeds c. Next is the "photosphere". That's where the escape speed is almost exactly c. At that radius, photons go into orbit around the hole and remain there for all time. There's the "ergosphere", a range in which energy can be extracted from the hole. A radius called the "static limit" is also involved. That's the distance at which it is impossible to stop moving, although that movement can include enough tangental speed to escape from the hole.
Sorry, but your question was a bit more vague than you probably intended.

 

[hasan_researc]

Thank you for the reply.

I am wondering why a star has to be of at least 3.2 solar masses in order to collapse into a black hole. In other words, how I could show that that's a trick question?

 

[Danger]

It's not impossible that a black hole of one solar mass could exist, but it could not be a consequence of natural stellar evolution.
The "why" involves really complicated aspects of nuclear fusion processes. I don't pretend to understand more than a small percentage of it. Basically, though, every time that the fusion stage of a star progresses, the energy output increases and the stellar envelope undergoes a brief ballooning followed by a shrinkage. Ie: when one fusion process runs out, the energy pressure can no longer support the stellar material against gravity. The matter falls in violently, triggering the next stage of fusion which then re-expands the star. Initially, hydrogen fuses into helium. In the next stage, helium-3 fuses into helium-4 and lithium. Subsequent fusion reactions produce carbon, oxygen, etc. until it gets to iron. Iron absolutely will not not fuse. When it reaches that stage, the fusion "fire" is extinguished. In a star of Sol's mass, the infalling matter then undergoes a final "fusion bounce" and puffs up into a red giant. It's a lot hotter than the original star, but the mass is distributed over such a broad volume that the average temperature is fairly low. The normal intra-atomic electronic repulsion (called "degenerate electron pressure") is enough to keep things in a normal realm.
When more mass is left in the core, the electronic repulsion is no longer strong enough to overcome gravity. Electrons get compressed into the nuclei of the atoms, to combine with the protons and become neutrons. The resulting "neutronium" is the densest possible material in the universe, and is the material of which neutron stars are composed. (Also George Bush's brain, but that's a subject for the Political Science forum...)
When the remaining mass of a star is sufficient, neither degenerate electron pressure nor any other force is sufficient to counteract gravity, so it continues contracting in size until it disappears. While the physical object is no longer noticeably present, the gravitational (and electric) fields still are.
This is really getting to the point where this thread might best be moved to the Astrophysics sub-forum. I'm getting way over my head here.

 

[hasan_researc]

Actually I am a first year physics undegrad. And that question was part of my "Professional Skills Problem Solving" exercise.

The exercise is meant to help me develop my skills in using the concept of energy and idealised models to arrive at an order of magnitude estimate of the target variable.

That's all I have to do, so I am puzzled as to why someone would set such an exercise in the first year problem solving exercise.

 

[Danger]

I am puzzled as to why someone would set such an exercise in the first year problem solving exercise.

Beats me; I never finished high-school. It might be worth your while to ask your instructor what the purpose is. Even if it isn't appreciated, it's proof of thoroughness, which is essential to science.

 

[Sakha]

You can use the Swarzchild (probably mispelled) radius. I think I've read somewhere that it would be 18.5km. Deriving Swarizchild radius in a classical way is pretty easy I think, just setting the escape velocity at the velocity of light.