Astronomy
# Life Cycles of Stars

Approximately \(4.6\) billion years ago, the Sun formed from of a huge cloud of **collapsing gas and dust**. The action of gravity pulled the mass of the cloud into a much denser body of mostly hydrogen and helium. The continual contraction coincided with an increasing rate of spin, which caused the heavier elements to be ejected, leaving us with the Sun we know and love.

Our objective in this quiz is to explore what factors affect whether a vast region of gas and dust (known as a nebula) will coalesce to form a star, or quietly diffuse into the vacuum of space.

Before the Sun was at the center of our solar system, its mass was spread out over a large region of space extending beyond the orbit of Neptune. Our current hypothesis of solar system formation posits that, over several hundred million years, this gas collapsed as a result of gravity acting between matter particles.

**Gravity** is stronger between masses that are closer together. Compare two nebulae that have the same radius \(R\) but different densities. In which nebula are individual gas molecules **closer** on average (and consequently feel a **larger** attractive force toward their neighbors)?

**Assume** that nebula A is denser than nebula B.

If the Sun's mass is scattered uniformly over the solar system, the density of hydrogen gas in this region would be about \(\SI[per-mode=symbol]{1e-9}{\kilo\gram\per\meter\cubed}.\)

At this density, two gas molecules with mass \(\SI{1e-27}{\kilo\gram}\) each are separated on average by a distance of \(\SI{1e-6}{\meter}\).

What is the approximate magnitude of the force of gravity between two adjacent gas molecules?

**Assume** that the gravitational constant is \(G=\SI[per-mode=symbol]{6.67e-11}{\meter\cubed\per\kilo\gram\per\second\squared}.\)

Gas and dust particles in a nebula are drawn together by the force of gravity, but not without any resistance. In our quiz on **matter**, we introduced the idea of gas pressure, an outward-directed force due to the thermal motion of molecules.

Suppose two nebulae are identical in size and density, but nebula **B** has a higher temperature than nebula **A**. In which nebula is the outward gas pressure opposing collapse **greater**?

**Assume** that temperature is uniform through the bulk of both nebulae.

We have seen that if the nebula is going to collapse, gravity's inward pull must win over the outward push of gas pressure. In the remainder of this quiz, we are going to discover a simple condition on the **total energy** (potential energy plus kinetic energy) of the nebula to serve as a test of whether a star will coalesce.

Gas temperature is really a measure of how much energy its molecules have on average, but for historical and practical reasons, we measure temperature in Kelvin and not Joules. Boltzmann's constant \(k_B =\SI[per-mode=symbol]{1.381e-23}{\joule\per\kelvin}\) is the proportionality constant between energy and temperature, which is evident in its unit: Joules per Kelvin.

Suppose we have a sample of \(N\) hydrogen molecules from a nebula in a box. The total energy is the sum of the energies of all \(N\) molecules. Based on units, which expression would you expect is characteristic of the **total energy** within the box when it has temperature \(T?\)

If we were very careful in modeling our ideal gas, we would find that the proportionality between \(E_\text{tot}\) and \(T\) is \[E_\text{tot}=\frac32 N k_B T.\] This is the

total kinetic energyof \(N\) gas molecules at temperature \(T.\)

In the absence of gravity, the thermal motion of gas molecules would lead to the dispersal of the nebula until its density eventually approaches \(0\). If its total mass is large enough, gravitational forces will prevent diffusion. Therefore, we expect that for a nebula with a particular radius and temperature, there is a **minimal mass** of gas that will eventually be drawn together by gravity. In the problems to follow, we will compute this mass, called the **Jeans mass**.

The gravitational potential energy of a gas molecule is related to the amount of energy we would need to spend to remove it from the nebula. We learned that the gravitational potential energy of a gas molecule with mass \(m\) near a large, roughly spherical nebula with mass \(M\) is \[U_g = -\frac{GMm}{R},\] where \(R\) is the radius of the nebula.

What is the **potential energy** of a hydrogen atom at the edge of the nebula?

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**Details and Assumptions:**

- Gravitational potential energy is conventionally negative to remind us that gravity binds mass together.
- The mass of a hydrogen atom is \(\SI{1.66e-27}{\kilo\gram}.\)
- Since we are interested in the conditions from which the Sun formed, assume \(M\) is equal to the mass of the Sun \(\SI{1.99e30}{\kilo\gram}.\)
- The radius of the nebula is about \(\SI{50}{au}=\SI{7.5e9}{\kilo\meter}.\)

Each gas atom has kinetic energy \(K\) depending on its speed, so its total energy \((E=K+U)\) can be negative or positive since \(U\) is negative and \(K\) is positive. If \(K\) is large enough, and it doesn't collide with anything, it can leave the gravitational pull of the rest of the nebula. In fact, the overall sign of the energy lets us decide the ultimate fate of the atom.

If a gas atom (with mass \(m\)) on the edge of the nebula \((\)with mass \(M)\) takes off on a trajectory that takes it away from the rest of the gas atoms, how much energy must it have to get away and never come back?

We would like to know whether the entire nebula holds together; you have calculated the potential energy of just one gas molecule. Each gas molecule contributes some potential energy to the total potential energy of the nebula, which is related to the energy needed to pull the nebula completely apart.

Estimate the total number \(N\) of gas molecules in the nebula, and use the result of the previous question to compute the order of magnitude of the gravitational potential energy of a spherical mass of gas that has mass \(M\) and radius \(R\).

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**Details and Assumptions:**

- Assume the individual gas molecules are all identical and have a mass of \(m\) each.
- Not all of the gas molecules have the same potential energy, but in order to estimate the total gravitational potential energy of the nebula, assume the result of the previous question is roughly the potential energy per molecule.

When a nebula eventually drifts apart without producing stars, each atom has enough energy to overcome the gravity of all other atoms. We can use this principle to estimate the minimal mass of the nebula that gave birth to the Sun.

Because of spectral measurements, astronomers can tell the typical temperature of a nebula; for example, the one in which our Sun formed is about \(\SI{5000}{\kelvin}.\) Estimate the mass of the smallest star that can form from a nebula with this temperature and a radius of \(\SI{50}{au}.\)

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**Details and Assumptions:**

- Assume the nebula is roughly spherical and made of hydrogen gas. A hydrogen atom has mass \(m=\SI{1.66e-27}{\kilo\gram}.\)
- \(\SI{1}{au}=\SI{1.5e8}{\kilo\meter}.\)
- Recall that the total kinetic energy of \(N\) ideal gas molecules with temperature \(T\) is \(\frac{3}{2}Nk_B T,\) where \(k_B= \SI[per-mode=symbol]{1.381e-23}{\joule\per\kelvin}.\)

Large, massive nebulae (informally called star sanctuaries) are common in the spiral arms of the Milky Way. In this quiz, we have found a rough criterion for the critical mass (the **Jeans mass**) of a nebula that will produce new stars.

Instead of analyzing specific forces on the gas, we considered the total kinetic energy of gas molecules and balanced it against the total gravitational potential energy. This allowed us to sweep the detailed balance of forces under the table. We explore at length in the next quiz the exact details behind our approach pitting gas pressure against gravitational forces.

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