Determine the radius of a neutron star using the same argument...

Question

Determine the radius of a neutron star using the same argument as in Problem 65 but for N neutrons only. Show that the radius of a neutron star, of 1.5 solar masses, is about 11 km.

Accepted Answer

Hello, fellow physicists today or solved the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A physicist estimates the radius of a neutron star that has a mass of 2.0 solar masses. Assuming that her assumptions are reasonable. What value for the radius does she come up with hint the solar mass is approximately 2.0 multiplied by 10 to the power of 30 kg. Awesome. So it appears for this particular problem. The final answer that we're ultimately trying to solve for is we're trying to figure out what radius value will this physicist come up with using the provided information that's given to us by the prom itself? So now that we know that we're trying to figure out what their radius value is. Let's read off our multiple choice answers to see what our final answer might be noting that they're all in the same units of kilometers. So A is 2.4 B is 4.6 C is 7.3 and D is 9.9. OK. So first off, we should recall a note that we know if all the energy levels are filling up to a Fermi energy, which is denoted as ef the average energy will be given by 3/5 multiplied by ef which once again, ef is our Fermi energy. So therefore, we can go ahead and write that the energy due to the energy levels are so we can easily write that E is equal to N multiplied by 3/5 multiplied by ef which once again, ef is our Fermi energy fantastic, which we can go ahead and also write that E is equal to N multiplied by 3/5 which is multiplied by H squared divided by eight, multiplied by M all multiplied by three divided by pi multiplied by N divided by V. Awesome. And then this is all to the power of two, divided by three fantastic. Where capital N in this case is the number of neutrons within the neutron star. M is the mass of a single neutron and V is the volume of the neutron star. So we should not go take a moment here to pause and recall and note that the entire mass is due to the neutrons. Therefore, the mass of a neutron star, which we're gonna call the mass of a neutron star. M is equal to N multiplied by lowercase M. Thus, we could go ahead and write that E is equal to capital M divided by lower case M multiplied by 3/5 multiplied by H squared, divided by eight, multiplied by M all multiplied by three, divided by pi multiplied by capital M divided by four, divided by three multiplied by pi multiplied by R to the power of three multiplied by M. And this whole expression is to the power of two thirds and are four divided by three multiplied by pi multiplied by R to the power of three. So that is considering that the neutron star is spherical, OK, moving right along here and R in this case, lowercase R is the radius of the neutron star. So when we simplify this equation, we will find that E is equal to three multiplied by 18. And this is gonna be to the power of two thirds multiplied by H squared multiplied by capital M to the power of five thirds, all divided by 160 multiplied by pi to the power of four thirds multiplied by M to the power of eight thirds multiplied by R to the power of two fantastic. So now at this point, as we should recall and note, and as we should know, the gravitational potential energy for a uniform sphere of mass M and radius R is given by the following equation which as we should recall, as U is equal to negative 3/5 multiplied by capital G multiplied by capital M squared divided by R where capital G is our gravitational constant. So therefore, considering this the total energy which we call E capital T. So E total is equal to three multiplied by 18 to the power of two thirds multiplied by H squared, multiplied by capital M to the power of five thirds divided by 160 multiplied by pi to the power of four thirds multiplied by M to the power of eight thirds multiplied by R to the power of two minus three divided by five, multiplied by capital G multiplied by capital M squared divided by R Awesome. So now let us take the constant P and let's write this as in blue. So the constant P is equal to three multiplied by 18 to the power of three halves multiplied by H squared multiplied by capital M to the power of five thirds divided by 160 multiplied by pi to the power of four thirds multiplied by M to the power of of eight third. And let us also say that Q is equal to three divided by five, multiplied by capital G multiplied by capital M squared. Awesome. So when we do that, we will find that E total is equal to P divided by R squared minus Q divided by, so Q divided by R Fantastic. So that's when we're using our new constants that we defined up above. OK. So with that in mind, so now we need to find the radius. So in order to find the radius, we need to set the derivative of E total is equal to zero. So let's make a little note here. So when E T equals zero, since we're sending the derivative E total equals zero, meaning we could go ahead and write that D ET by Dr is equal to zero. So let me do that, that will give us that R is equal to. So R the, which once again our radius is equal to two multiplied by P divided by Q. Awesome. So then therefore, when we plug in our constant, we will get that R is equal to two, multiplied by three, multiplied by 18 to the power of two thirds. Uh So it's 18 to the power of two thirds multiplied by H squared multiplied by capital M to the power of five thirds. Fantastic. Divided by 160 multiplied by pi to the power of four thirds. And then we need to multiply this value by lower case M to the power of eight thirds. And then we need to divide this by 3/5 multiplied by capital G multiplied by capital M to the power of two Fantastic. So ultimately, this expression will simplify to R equals 18 to the power of so we take 18 to the power of two thirds. Awesome. So 18 to the power of two thirds. And then we need to multiply this value by H squared multiplied by capital or so. And then by H squared means the MS, the M will move down below. So once again, so R is equal to 18 to the power of two thirds multiplied by H to the power of two, all divided by 16. Once again, this is because we're simplifying 16 multiplied by pi to the power of four thirds. So four thirds, and then we need to multiply this by capital G multiply this by capital M to the power of one third. And I mean to multiply this by lowercase M which is to the power of eight thirds. Fantastic. So now all we need to do is substitute in all of our known variables and sulfur R. So let us quickly note here off to the side that our value for H is going to be 6.63 multiplied by 10 to the power of negative 34. And its units of measurement are going to be joules multiplied by seconds. Awesome. So let's make a. So once again, I'm gonna make a little star here. So this is H, our value for H equals that. And then our value for G as we should recall, which is our gravitational constant is equal to 6.67 multiplied by 10 to the power of negative 11. And don't forget that our units of measurement are going to be newtons multiplied by meters square divided by kilograms squared. Awesome. And then we should recall and note that our value for capital M is going to be equal to two 0.0 multiplied by. So it's gonna be 2.0 multiplied by 2.0 multiplied by 10 to the power of drum rolls 30 so 10 to the power of 30. And then don't forget that it's in units of kilograms. And then lowercase M is gonna be equal to 1.67 multiplied by 10 to the power of negative 27 kilograms. Awesome. So when you plug in these known variables into our equation to solve for R, you will find that R when we round to two decimal places and are round and making sure our answer is converted to kilometers, we will find that our final answer is approximately equal to 9.9 kilometers and that is our final answer. All right, we did it. So let's quickly note here that initially when you type this into your calculator, you will find that R our radius is equal to nine. So it's equal to 9866. So 9866 point 5 6 1 m and you need to convert units of meters to units of kilometers. So they have to make sure that this is in units of meters. And that's why you have to convert meters to kilometers because all of our final answers are in units of kilometers. So that's it. So going back to the top, let's see what our final answer has to be. It. Appears. The final answer has to be the letter D which is 9.9 kilometers. Thank you so much for watching. Hopefully, that helped and I can't wait to see.

Determine the radius of a neutron star using the same argument as in Problem 65 but for N neutrons only. Show that the radius of a neutron star, of 1.5 solar masses, is about 11 km.

Key Concepts

Neutron Star Structure

Gravitational Binding Energy

Mass-Radius Relationship

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