What happens if you keep adding neutrons—forever? Is there a limit? And what does quantum mechanics and relativity have to do with it?
*https://www.youtube.com/watch?v=LEyS7-OK57I
**https://300.ya.ru/v_iITuBqIU
таймкоды
00:00:00 Введение в тему нейтронов
- Обсуждение возможности создания объектов из нейтронов.
- Упоминание о тонких и удивительных аспектах квантовой механики, ядерных взаимодействий и гравитации.
00:00:39 Что такое нейтрон?
- Нейтроны — нейтральные частицы в атомных ядрах.
- Они не отталкиваются и не притягиваются друг к другу из-за отсутствия электрического заряда.
00:01:38 Роль нейтронов в ядре
- Нейтроны поддерживают целостность ядра.
- Протоны отталкиваются друг от друга из-за положительного заряда.
00:01:58 Сила отталкивания протонов
- Расчёт силы отталкивания между двумя протонами с помощью закона Кулона.
- Сила отталкивания эквивалентна весу 23-килограммового объекта.
00:03:25 Гравитационное притяжение протонов
- Оценка гравитационного притяжения между двумя протонами.
- Гравитационная сила слишком мала для удержания ядра.
00:05:13 Сильное ядерное взаимодействие
- Сильное ядерное взаимодействие удерживает нуклоны вместе.
- Оно действует только на коротких расстояниях.
00:07:19 Влияние нейтронов на стабильность ядер
- Нейтроны помогают уравновесить отталкивание протонов.
- Без нейтронов ядра стали бы нестабильными.
00:08:27 Ограничения квантовой механики
- Принцип исключения Паули ограничивает количество нейтронов в ядре.
- Увеличение энергии нейтронов приводит к их улету.
00:09:46 Возможность гравитационного удержания нейтронов
- Вопрос о том, насколько массивной должна быть совокупность нейтронов для гравитационного удержания.
- Использование аргумента Бернарда Шутса для оценки.
00:11:01 Оценка энергии связи нейтронов
- Экспериментально доказано, что для удаления нуклона из ядра требуется около 8 МэВ энергии.
- Преобразование энергии связи в космическую скорость для оценки возможности гравитационного удержания.
00:11:35 Оценка скорости убегания нейтрона
- Скорость убегания нейтрона с энергией 8 МэВ составляет около 3,92 × 10^7 м/с, что более чем на 10% превышает скорость света.
- Возникает вопрос: какой массы должен быть объект, состоящий из нейтронов, чтобы его гравитационная скорость убегания была такой высокой?
00:12:28 Определение нейтронной звезды
- Нейтронная звезда — это объект, состоящий почти полностью из нейтронов, удерживаемый вместе гравитацией, а не сильным ядерным взаимодействием.
- Гравитационное поле нейтронной звезды настолько сильное, что даже нейтроны с кинетической энергией 8 МэВ не могут вырваться наружу.
00:13:15 Оценка минимальной массы нейтронной звезды
- Минимальная масса нейтронной звезды оценивается с помощью ньютоновской физики.
- Для сферического объекта массой m и радиусом r скорость отрыва от поверхности равна квадратному корню из двух gm/r.
- Плотность объекта, состоящего из нейтронов, равна массе, делённой на объём, что позволяет использовать уравнение плотности.
00:14:26 Зависимость скорости убегания от массы и плотности
- Скорость убегания зависит от третьей степени массы объекта.
- Чем массивнее объект, тем выше его скорость убегания из-за большего гравитационного притяжения.
00:15:12 Первая оценка минимальной массы
- Минимальная масса, необходимая для удержания нейтронов гравитацией, оценивается как 3,9 × 10^28 кг.
- Эта оценка составляет примерно 0,2 массы Солнца.
00:16:46 Второй ньютоновский подход
- Второй подход учитывает общую энергию гравитационного взаимодействия.
- Суммарная энергия гравитационного взаимодействия определяется как 3/5 gm^2/r.
- Общая энергия ядерного взаимодействия оценивается как произведение количества нейтронов на энергию связи одного нейтрона.
00:20:29 Новая оценка минимальной массы
- Новая оценка минимальной массы составляет 8,5 × 10^28 кг, что более чем в два раза превышает предыдущую оценку.
- Этот подход учитывает общую энергию, необходимую для освобождения всего объекта, а не только поверхностные нейтроны.
00:21:33 Роль общей теории относительности
- Внутри нейтронной звезды гравитация настолько сильна, что давление вносит свой вклад в гравитационное поле.
- Уравнения Толмана-Оппенгеймера-Волкова описывают релятивистскую версию гидростатического равновесия.
- Истинная минимальная масса стабильной нейтронной звезды приближается к 0,1 массы Солнца.
00:22:57 Заключение
- Подчёркивается близость ньютоновской оценки к истинной минимальной массе нейтронной звезды.
- Обсуждается возможность исследования экстремальных объектов во Вселенной с помощью простых математических инструментов.
00:23:13 Предел массы нейтронных звёзд
- Гравитация притягивает внутрь, а давление нейтронного вырождения выталкивает наружу.
- Нейтроны, как и электроны, являются фермионами, и принцип исключения Паули запрещает им занимать одно и то же квантовое состояние.
- При увеличении массы гравитация преодолевает давление нейтронного вырождения, и звезда превращается в чёрную дыру.
- Предел Толмана-Оппенгеймера-Волкова находится между 2/3 и 2,9 массами Солнца.
00:24:17 Наблюдаемые нейтронные звёзды
- Большинство нейтронных звёзд имеют массу около 1,2–2,1 массы Солнца, в среднем около 1,4 массы Солнца.
- Их радиус составляет от 11 до 13 километров, что сравнимо с шириной Манхэттена.
00:24:55 Плотность и радиус нейтронных звёзд
- Плотность нейтронных звёзд варьируется от 2,1 до 17 кг/м³.
- Ядра нейтронных звёзд намного плотнее внешних слоёв.
- Простая оценка радиуса оказывается удивительно точной.
00:26:22 Скорость вращения нейтронных звёзд
- Скорость отрыва от поверхности нейтронной звезды составляет более половины скорости света.
- Теоретически нейтронная звезда может вращаться со скоростью около тысячи оборотов в секунду.
- Пульсары — это нейтронные звёзды, вращающиеся сотни раз в секунду и распространяющие высокоэнергетическое излучение.
00:27:39 Образование нейтронных звёзд
- Нейтронные звёзды образуются в результате катастрофической гибели массивных звёзд при взрывах сверхновых.
- Если масса коллапсирующего ядра превышает предел Чандрасекара, электроны сталкиваются с протонами, создавая нейтроны и высвобождая поток нейтрино.
- Внезапное уплотнение останавливает коллапс, образуя протонно-нейтронную звезду.
00:28:37 Роль нейтронных звёзд в формировании элементов
- Давление нейтронного вырождения вызывает отскок во внутреннем ядре и запускает ударную волну наружу.
- Огромное количество вылетающих нейтрино накапливает энергию во внешних слоях, усиливая удар и приводя к коллапсу ядра сверхновой.
- Взрыв рассеивает по галактике элементы, составляющие планеты и жизнь.
00:29:14 Значение нейтронных звёзд для эволюции Вселенной
- Если бы максимальная масса нейтронной звезды была меньше, коллапсирующие ядра продолжали бы движение в чёрные дыры.
- Отсутствие сверхновых означало бы отсутствие рассеяния тяжёлых элементов, планет, химии и жизни.
- Нейтронные звёзды — важные участники космической истории, связанные с экстремальными процессами во Вселенной.
00:29:58 Заключение
- Нейтронные звёзды — это экзотические остатки звёздного коллапса, играющие ключевую роль в формировании элементов и планет.
- Наше существование связано с экстремальными процессами во Вселенной.
Transcript
0:00
Have you ever wondered what would happen if you built an object made entirely of neutrons? Those tiny neutral particles
0:08
tucked inside atomic nuclei. For example, could you build a neutron
0:13
tennis ball, a neutron football, or a mountain made of nothing but neutrons?
0:19
And could you keep going making ever larger objects? Or is there a limit? Well, as is often the case in physics,
0:26
it turns out that the answer is equal parts subtle and marvelous.
0:32
And understanding why takes us deep into the heart of quantum mechanics, nuclear
0:37
forces, and gravity. This is a story where imagination and
0:42
reality collide. Where spaceime bends and the familiar laws of physics are
0:47
stretched to their limits. Where quantum principles clash with gravity on a cosmic scale and something extraordinary
0:55
emerges from the wreckage. So, if you’re ready, buckle up and enjoy the ride.
1:01
Our story begins with a simple question. What exactly is a neutron? Well, as you
1:07
probably know, at the center of an atom lies the atomic nucleus. Neutrons are
1:12
tiny neutrally charged particles that are found inside the nucleus, right alongside positively charged protons.
1:20
And because neutrons carry no electric charge, they don’t electrically repel or
1:25
attract one another. They don’t even electrically push or pull on the protons that they sit next to.
1:31
But if neutrons don’t electrically push or pull on anything, then what’s their purpose? What do they actually do? Well,
1:38
it turns out they play a vital role in holding the nucleus together. And this will be crucial to our story. To
1:44
understand why, we need to first consider what’s happening with the protons inside the nucleus. And unlike
1:50
neutrons, protons are positively charged, which means they repel each other constantly.
1:57
Imagine two protons sitting one phento meter apart. That’s a typical separation
2:03
inside a nucleus equal to about 1 * 10 -15 m. We can calculate the repulsive
2:09
force between these two protons using Kulum’s law which states that the force between two charges Q1 and Q2 is
2:17
proportional to the product of the charges and inversely proportional to the square of the distance between them
2:23
R 2. Here epsilon not is the permitivity of free space. In the case of the two
2:29
protons both q1 and q2 are equal to e which is the fundamental unit of charge.
2:35
And therefore our force equation takes the following form. And if we then plug in the numbers for the charge and
2:41
distance we find a force of roughly 230 ntons.
2:46
Now this might not sound like a very large force but remember that this force is acting at a scale of 1 phtometer
2:54
between two tiny little protons. It’s immense. 230 ntons is equivalent to the
3:01
weight of a 23 kg object on Earth. For example, imagine holding a fully loaded
3:07
large suitcase ready for international travel at arms length. So when two
3:13
protons are just one phento meter apart, they’re pushing on each other with the same force it takes to hold up that
3:20
suitcase with one arm. That is a colossal repulsive force. So given the
3:26
strength of repulsion between protons, why don’t atomic nuclei just fly apart?
3:32
There must be something holding them together. Maybe it’s gravity. After all,
3:38
gravity is an attractive force that acts between any objects with mass and protons and neutrons both have mass. So,
3:45
let’s check whether this is possible. If we again consider our two protons
3:50
separated by a distance of one phentometer, then we can use Newton’s famous universal law of gravitation to
3:56
estimate the gravitational attraction between the two protons. In this equation, m1 and m2 refer to the masses
4:02
of our two objects. R represents the separation and G is Newton’s gravitational constant. In the case of
4:09
two protons, both have a mass of approximately 1.67 * 10 -27 kg. And so
4:16
if we plug in the numbers, we find a gravitational force of roughly 1.87 * 10
4:22
34 newtons. Now that’s not just small, it’s utterly negligible.
4:29
If we compare with the electrical repulsive force between the two protons that we calculated earlier and calculate
4:35
the ratio of electrostatic to gravitational force, we see that the electrostatic repulsive force is greater
4:41
than the gravitational force by a factor of more than 10 ^ of 36. If that doesn’t
4:47
immediately whack you in the face, then try writing out the number by hand. So,
4:53
it’s clearly not gravity holding the nucleus together. is just far far too
4:58
weak. And this brings us back to the neutrons because there is a force strong enough to overcome all that repulsion. A
5:06
force that acts between all nucleons, protons and protons, protons and neutrons, neutrons and neutrons. And
5:13
it’s called the strong nuclear force. And it’s around 100 times stronger than the electromagnetic force, but only at
5:21
extremely short distances. Unlike gravity or electromagnetism which act over long ranges, the strong force is
5:28
strictly local. It only operates between immediate neighbors. And this limitation
5:34
has important consequences. Let’s consider a large nucleus where the
5:39
red spheres represent protons and the blue spheres represent neutrons. If we then focus in on one of the protons,
5:46
then due to the short range nature of the strong nuclear force, this proton will only be significantly affected by
5:53
protons and neutrons that are neighboring it. And this can be seen clearly in the diagram where those
5:59
nucleons surrounding the proton have been highlighted and the proton is only tightly bound by these nucleons.
6:06
Likewise, if we focus on a different proton, it too will only be tightly held by its neighboring protons and neutrons.
6:14
And the same is true for any proton that we zoom in on. It only experiences a
6:19
significant binding from the strong force due to its immediate neighbors. Contrast that with the electrostatic
6:26
repulsive force, which is a long range force, meaning that two protons on the opposite side of the nucleus can still
6:34
exert a repulsive influence on each other. And this is a crucial point because if we consider a single proton,
6:42
then this proton will only be bound by the strong nuclear force due to the surrounding nucleons.
6:49
Whereas that same proton will experience a repulsive force from all the other
6:54
protons within the nucleus and that repulsive force scales with the number of protons while the binding from the
7:00
strong nuclear force stays limited or even diminishes due to geometric crowding. Eventually the repulsion
7:07
becomes too strong and the nucleus tips into instability. That’s why large
7:13
elements like uranium are radioactive. They’re right on the edge of falling apart.
7:19
And this is where neutrons play a remarkable role. Because neutrons are electrically neutral, they don’t
7:25
contribute to the repulsion at all, no matter how many you add. But they do contribute to the strong nuclear force.
7:33
They help glue the nucleus together without increasing the internal stress. In that sense, neutrons are incredibly
7:40
efficient. They add to the binding energy without adding to the electrostatic cost. As nuclei grow
7:47
larger, more and more neutrons are needed to counterbalance the growing proton repulsion. Without them, the
7:53
nuclei would simply tear itself apart. Neutrons quietly and invisibly are what
7:59
make large atoms possible. So, it would seem that these magical
8:04
neutrons are the perfect building blocks. You can just keep adding them and they strengthen the nucleus via the
8:10
strong nuclear force without contributing any repulsive push back. No charge, no conflict, just pure binding.
8:18
It starts to look like there’s no limit to how large an object you could build just from neutrons alone. They seem
8:25
perfect for the job. But this is where quantum mechanics slams the door shut.
8:31
You see, neutrons are firmians, particles named after the Italian physicist Enrio Fermy. And the thing
8:39
about firmians is that no two firmians can occupy the same quantum state at the
8:44
same time. This rule is known as the ply exclusion principle.
8:50
Now when we apply this to a nucleus, it means neutrons are forced to occupy distinct energy levels. They can’t all
8:56
just pile into the lowest one. In fact, you can fit two neutrons into that lowest level. One with spin up and one
9:04
with spin down. but the next pair must move to the next highest level and so on
9:09
and so forth. It’s a bit like hotel guests filling the lowest floors first, then moving up as rooms run out. As you
9:16
keep adding neutrons, the lower levels fill up and new arrivals are pushed into higher and higher energy states. Their
9:24
average energy increases and eventually it becomes too much. Once that energy
9:29
exceeds the binding energy holding them in place, the strong force can no longer contain them and neutrons start to
9:36
escape. So in a very real sense, it appears as if quantum mechanics builds
9:41
in a limit to how many neutrons you can keep piling together. To summarize, when dealing with a
9:48
nucleus, too many protons and electrostatic repulsion dominates, but too many neutrons and power’s exclusion
9:55
principle drives up the energy too high. But here’s the twist. What if we didn’t
10:01
rely on the strong force to hold neutrons together? What if we packed so many of them into a single object at
10:08
nuclear densities that gravity itself finally got strong enough to take over?
10:14
That might sound implausible given that we’ve already seen how weak gravity is compared to other fundamental forces.
10:21
But if we could gather enough neutrons together, then perhaps their combined mass could begin to make gravity matter.
10:29
So the key question becomes, how massive would a collection of neutrons need to be at nuclear density for gravity alone
10:38
to keep them bound together? To get an initial estimate, we’ll use a
10:43
beautifully simple back of the envelope argument due to Bernard Schutz from his book, Gravity from the Ground Up, that
10:49
only requires high school mathematics. It’s a remarkably effective way to cut through the complexity and arrive at an
10:56
answer that’s strikingly close to what general relativity predicts.
11:01
Let’s begin with something that is experimentally very well established. In a typical atomic nucleus, it takes about
11:08
eight mega electron volts of energy to remove a nucleon, whether a proton or a
11:14
neutron. This is the average binding energy per nucleon. Now, here’s the
11:19
clever step. We can convert that escape energy into an escape velocity. To do
11:26
this, we assume that a neutron would need this much kinetic energy to escape. And we then apply the classical kinetic
11:33
energy equation/ mv^ 2. And if we then rearrange this for velocity, we find
11:38
that the escape velocity will be equal to the square of 2 e / m.
11:45
Now an energy of 8 mega electron volt is equivalent to about 1.28 * 10 -12 jw.
11:52
And the mass of a neutron is roughly equal to 1.67 * 10 -27 kg. If we plug
11:59
those values into our escape velocity equation, the equation gives a value of 3.92 * 10 7 m/s. That’s over 10% the
12:08
speed of light. Okay, so now we have an estimate for the escape velocity required. So that leads to a natural
12:15
question. How massive would an object made entirely of neutrons packed at nuclear
12:22
density need to be so that its gravitational escape velocity is this high?
12:28
Or to put it another way, what mass would create such strong gravitational binding so that even a neutron with 8
12:35
mega electron volts of kinetic energy, enough to escape a typical nucleus, would still be trapped. If such an
12:42
object exists, it would no longer rely on the strong nuclear force to stay together. It would now be bound together
12:49
by gravity. And that fundamentally is what defines a
12:55
neutron star. A stellar corpse so dense, so massive that even individual neutrons
13:01
are locked in place by gravity alone. An object made almost entirely of neutrons,
13:07
held together not by the strong force, but by the crushing grip of its own gravitational field.
13:15
And here’s the astonishing part. We can estimate the minimum mass needed to create such an object using nothing more
13:22
than Newtonian physics. And the strategy is simple. As we’ve already established, such an object must
13:30
be massive enough at nuclear density such that its escape velocity is greater
13:35
than about 3.9 * 10 7 m/s. The velocity a neutron would need to escape if it had
13:42
8 mega electron volts of kinetic energy. Now for a spherical object of mass m and
13:48
radius r, the escape velocity from the surface is equal to the square roo of 2 gm / r, where g is Newton’s
13:56
gravitational constant. Because we’re imagining the object is made entirely of neutrons packed at nuclear density, we
14:04
can use the equation for density, which is simply equal to the mass divided by the volume. And assuming a sphere of
14:11
radius r, the volume of the sphere is equal to 4/3 p r cubed. And so our
14:17
density equation takes the following form. If we then rearrange this expression for the radius, we find the
14:24
following result. The next step is to substitute this expression for r back into the escape
14:31
velocity equation and we find the following result. Then after a bit of simplifying and combining of terms, we
14:38
end up with the following simplified expression. Now this is a powerful result. It tells us how the escape
14:46
velocity of a neutron star depends on its mass and its density. And we’re assuming that density is constant and
14:52
equal to the typical nuclear value. And it makes sense for a constant density
14:58
object. We see that the escape velocity scales with the third power of mass. The
15:03
more massive an object, the higher the escape velocity, which makes sense since higher mass means higher gravitational
15:10
attraction at the surface. Okay, so we now want to use this equation to find out the mass that would
15:16
give an escape velocity of at least 3.92 * 10 7 m/s. That’s the speed we said a
15:23
neutron would need in order to escape. To do this, we rearrange our equation
15:28
for m. And if we then plug in the value for the escape velocity along with the typical density of a nucleus and
15:35
Newton’s gravitational constant, we find an estimate for the minimum mass of 3.9
15:41
10 28 kg. And if we compare this with the mass of
15:46
our sun, we see that this calculated minimum mass is approximately 0.02 * the
15:52
mass of the sun. And this gives us a first estimate for the minimum mass of a neutron star. the mass needed so that
15:59
gravity can prevent even the highest energy surface neutrons from escaping. If it’s any lighter, neutrons with 8
16:06
mega electron volts of kinetic energy could escape the gravitational pole. But
16:12
notice what this estimate is based on. It asks whether a single neutron at the
16:17
very edge has enough energy to escape. In that sense, it’s a surface-based
16:22
criterion, a local condition. It tells us the mass required to keep just the
16:28
outermost particles gravitationally bound. That’s useful and surprisingly effective, but it doesn’t tell the full
16:35
story. A neutron star isn’t held together just by clinging to its surface. It’s stable because every
16:41
neutron throughout the entire star is bound by the collective pull of gravity.
16:47
So to refine our estimate, we need to ask a deeper, more global question. how much gravitational energy is binding the
16:54
entire object together. This leads us to a second Newtonian approach, still a
17:00
back of the envelope estimate, but now one that considers the total gravitational binding energy. If that
17:06
energy exceeds the total binding energy the strong force would provide for the same mass of neutrons, then gravity has
17:13
truly taken over the job. So we’re still working within Newtonian physics, but
17:18
improving our estimate by shifting from a surface escape argument to a whole
17:23
object energy balance. So let’s see where this takes us. First, we need to
17:29
understand the idea of gravitational binding energy. This is the total energy
17:34
you’d have to supply to completely pull apart an object that’s held together by gravity. Not just a single particle, but
17:41
the entire system. For a spherical object of mass m and radius r, assuming uniform density, the
17:49
total gravitational binding energy is given by 35ths * g m^2 / r. This tells
17:56
us how much energy it takes to tear apart a self-gravitating object layer by
18:01
layer. And it can be calculated using simple integral calculus. The factor of
18:06
3 over5 isn’t arbitrary or a guess. It comes from integrating the gravitational
18:12
potential energy over all the concentric shells that make up the object.
18:18
Now, just like before, we’re assuming that the object is made of neutrons packed at nuclear density. So, we can
18:24
relate the mass and radius via the density equation. And if we write this in terms of the radius of the sphere and
18:31
then rearrange, we find the same expression that we derived earlier. Next, we can substitute this expression
18:38
for r back into our equation for the gravitational binding energy. And when we do that and simplify a bit, we find
18:45
the following beautiful equation. This gives us the total energy from gravity
18:51
that holds the object together. Assuming constant density, now let’s compare this to the total
18:58
nuclear binding energy of the same amount of matter. We know that in a typical atomic nucleus, each nucleon,
19:06
proton or neutron, is bound by around 8 mega electron volts. That’s the energy you’d need to pull a nucleon out of the
19:12
nucleus. So given that we’re assuming our object is made entirely of neutrons to estimate the total binding energy, we
19:20
simply need to multiply the average binding energy by the total number of neutrons. So how many neutrons do we
19:28
have? Well, if we assume our object has mass m and is made entirely of neutrons and if
19:35
the mass of a single neutron is labeled m subscript n, then the number of neutrons will be given by the following
19:42
equation. And if we label the binding energy per neutron as e subscript bind,
19:48
then the total nuclear binding energy can be estimated as the number of neutrons multiplied by the binding
19:55
energy per neutron. And if we sub in our expression for n, we find the following
20:00
relation. The next step is to set the gravitational binding energy that we
20:06
calculated earlier equal to the nuclear binding energy. And if we do that, we find the following result.
20:14
If we then rearrange and solve for m, then after a bit of work and simplification, we find the following
20:20
remarkably simple result as our new estimate for the minimum mass of our gravitationally bound neutron sphere.
20:29
And if we then plug in the numbers, we find an estimated minimum mass value of
20:35
8.5 * 10 28 kg. And this now corresponds to
20:42
approximately 0.0 043 solar masses, which is over twice our previous
20:47
estimate. And that makes sense. This approach considers the total energy needed to unbind the entire object, not
20:54
just whether one neutron at the surface can escape. It’s a more global and arguably more complete calculation.
21:01
Still, it’s a Newtonian estimate. And while our back of the envelope method brings us surprisingly close, it’s not
21:08
the full story. This is where general relativity becomes essential. Inside a neutron star, gravity is so intense that
21:16
even pressure contributes to the gravitational field. Spacetime itself curves inward, amplifying gravity’s
21:23
pull. These effects are captured by the Tolman Oppenheimer Vulov equations, which
21:29
describe the relativistic version of hydrostatic balance. At their core, they express how pressure, mass, and density
21:36
change with radius inside a spherically symmetric star. These coupled differential equations encode the curved
21:43
geometry of spacetime and the role of pressure as a source of gravity. Solving
21:49
them for a given equation of state, that is a relationship between pressure and
21:54
density, reveals the internal structure of a relativistic star like a neutron
21:59
star. And theoretical physicists have shown that by solving these equations, we find
22:06
that the true minimum mass for a stable neutron star is closer to about 0.1
22:11
solar masses. Slightly higher than our estimate, but still remarkably close
22:17
given the simplicity of our assumptions and our total neglect of relativistic effects.
22:23
And considering we use such a crude back of the envelope Newtonian approach stretching from the mass of a single
22:30
neutron around 10 — 27 kg all the way up to the mass of a neutron star about 10
22:37
28 kg that’s a range of more than 55 orders of magnitude to land within a
22:44
factor of 2 is pretty incredible. It shows just how powerful physical
22:49
intuition can be and how with nothing more than simple mathematical tools, we
22:54
can begin to explore some of the most extreme objects in the universe. Okay, so we’ve now estimated the minimum mass
23:01
required to gravitationally bind a collection of neutrons together. But what about the other end of the
23:08
scale? What’s the maximum mass a neutron star can have?
23:13
Well, that limit comes from a different kind of balance. Gravity pulling inwards versus neutron degeneracy pressure
23:20
pushing outwards. Neutrons like electrons are firmians. And as we’ve already noted, the ply exclusion
23:27
principle forbids them from occupying the same quantum state. When squeezed together, this generates an immense
23:33
outward pressure. But it has a limit. As the mass increases, gravity grows
23:38
stronger and eventually it overpowers even this neutron degeneracy pressure.
23:44
At this point, no stable configuration is possible. The star can no longer support itself and it will collapse into
23:51
a black hole. When general relativity is taken into account, this tipping point known as the
23:58
Tolman Oppenheimer Vulv limit is estimated to lie somewhere between 2.2 2
24:03
and 2.9 solar masses depending on the exact equation of state. In other words,
24:09
that’s the absolute upper mass limit a neutron star can have before collapsing
24:14
under its own gravity. But what do we see in reality? After
24:20
all, neutron stars aren’t just theoretical constructs. They’re actually out there, and we’ve observed plenty of
24:26
them. It turns out that the majority have masses between about 1.2 2 and 2.1
24:32
solar masses with the average clustering around 1.4 solar masses. That’s largely
24:37
a consequence of how they form. Something we’ll explore in a moment. But for now, let’s focus on this typical 1.4
24:45
solar mass neutron star. What would such a neutron star with an
24:50
average mass of 1.4 solar masses actually look like?
24:55
Well, if we assume it’s made of neutrons packed at constant density somewhere between 2 and 3 * 107 kg per cub m, then
25:04
we can estimate its radius using the equation we derived earlier. For a typical neutron star with a mass around
25:11
1.4 * that of the sun, this gives a radius in the range of about 13 to 15
25:17
km. In reality, neutron stars aren’t constant density spheres. Their cores
25:23
are far denser than their outer layers. And when we account for that using full general relativistic models, the
25:29
predicted radius comes down slightly to between 11 and 13 km. Even so, our
25:36
simple estimate gets surprisingly close. An entire star more massive than the sun
25:42
squeezed into a sphere no wider than a city. That’s truly mindblowing.
25:48
To emphasize this point, that’s comparable to the width of Manhattan in New York. An object more massive than
25:55
450,000 Earths crushed into a space no wider
26:01
than Manhattan. But to truly grasp how small this is, you have to zoom out. Manhattan is just
26:08
a dot on the map of the US. The US is just a patch on the surface of the Earth, and the Earth is a speck compared
26:14
to the Sun. Yet this tiny neutron star, no wider than a city, outweighs the
26:20
entire sun. And just a teaspoon of this matter would have a mass of about 1.4 trillion kg.
26:30
And assuming a typical radius of around 13 km, the escape velocity from the
26:36
surface would be roughly 1.67* 10 8 m/s, more than half the speed of light. A
26:43
satellite orbiting just above the surface would travel at around 1.18 * 10
26:48
8 m/s completing a full orbit in just 0.71 milliseconds.
26:55
And these numbers have consequences. They set physical limits on how fast the star can spin, for example, because if
27:02
it rotates any faster, then material at the equator would be flung off into space. So in theory, a neutron star
27:11
could spin up to around a thousand times per second without flying apart.
27:17
And indeed, we’ve observed such objects. They’re called pulsars, neutron stars
27:22
spinning hundreds of times per second, sweeping beams of high energy radiation across the cosmos like cosmic lighouses.
27:30
Their signals reach us with astonishing precision, ticking away like celestial clocks that we can detect right here on
27:38
Earth. And everything we’ve described, the mass limits, the size, the escape velocity,
27:44
the spin rate, all emerges from the remarkable interplay between quantum mechanics, nuclear physics, and general
27:51
relativity. And here’s the real twist. Neutron stars don’t form by gradually
27:57
piling up neutrons. Rather, they are born in the catastrophic deaths of massive stars in
28:04
cosmic supernova explosions. You see, at the end of their lives,
28:10
stars that are more massive than about eight solar masses exhaust their nuclear fuel and develop inert iron cores.
28:19
And if the mass of the collapsing core exceeds the Chandraar limit, which is about 1.4 four solar masses, then
28:27
electron degeneracy pressure is no longer sufficient to support it. Electrons are forced into protons,
28:33
creating neutrons and releasing a flood of neutrinos. This sudden stiffening
28:38
halts the collapse briefly due to neutron degeneracy pressure forming a proton neutron star.
28:46
That brief resistance is critical. It triggers a bounce in the inner core and launches a shock wave outwards. Although
28:54
this shock wave stalls, the vast number of escaping nutrinos deposit energy into the outer layers, reviving the shock and
29:01
leading to a core collapse supernova. This explosion is what disperses elements like carbon, oxygen, and iron
29:09
into the galaxy, the ingredients of planets and life.
29:14
But this only happens because the equations of state allows neutron stars
29:19
to exist with masses greater than the Chandra Secar limit. If the maximum mass
29:25
of a neutron star were lower, then collapsing cores above 1.4 solar masses
29:30
would continue directly into black holes. There would be no neutron star to create the bounce, no explosion, and no
29:37
scattering of heavy elements. In short, there would be no neutron stars, no core collapse supernova.
29:46
And no supernova means no dispersal of heavy elements. And no dispersal means no planets, no
29:54
chemistry, and ultimately no life. Neutron stars are more than just exotic
30:00
remnants of stellar collapse. They are essential actors in the cosmic story that led to us. Our very existence is
30:07
bound to the most extreme and violent processes in the universe. And somehow that’s beautiful.
30:14
So what happens if you keep adding neutrons? Well, stack enough and gravity takes over and you get a neutron star.
30:21
Stack too many and not even neutron degeneracy pressure can resist the crushing pull of gravity and a black
30:27
hole forms. But the incredible thing about our universe is that in the narrow window between too little and too much,
30:34
something remarkable happens. Stars explode, elements scatter, and planets
30:39
and people can form. So, thank you for watching, and until next time, goodbye.
30:47
And as always, a massive thank you to all my patrons, and a special shout out to the following who have been
30:53
incredibly generous with their support. Thank you so much. I couldn’t do it without