# Change in Radius of star during core contraction

• zsyed94
In summary, the conversation discusses estimating the expansion of a red giant's envelope based on conservation of energy. The process involves considering a star of mass M and initial radius R, with a contracting core of mass Mc and radius Rc. The phase being examined is one where there are no nuclear reactions in the core and hydrogen burning in the shell above is too slow to significantly contribute to the energy budget. The formula for gravitational potential energy is also mentioned. The attempt at a solution involves using the virial theorem and setting up an equation to solve for R1, but the answer is overly complicated and may have missed simplifications.
zsyed94

## Homework Statement

We can make a rough estimate of how much the envelope of a red giant should expand as a result of the contraction of its core based on conservation of energy. We will consider a star of mass M and initial radius R, with a core of mass Mc and radius Rc. We will focus on the phase when there are no nuclear reactions in the helium core, and when hydrogen burning in the shell above it either does not occur or occurs too slowly to make a significant contribution to the energy budget.

d. Now suppose that the core contracts from its initial radius Rc,0 to a smaller radius Rc,1. This causes the envelope to expand from its initial radius R0 to a new radius R1. Assuming that the total energy content of the star is conserved in the process, compute R1/R0 in terms of R0, Rc,0, Rc,1, Mc, and M. For numerical convenience, you may set all the α factors equal to 1.

## Homework Equations

E = Ω/2 = αGm^2/r, with m = Mc and r=Rc for the star's core, m = M-Mc and r = R for the envelope, assume R>>Rc.

Gravitational Potential Energy = Gm1m2/R

## The Attempt at a Solution

With conversation of energy, I tried

G(Mc)^2/Rc,0 + G(M-Mc)^2/R0 + GMc(M-Mc)/R0 = G(Mc)/Rc,1 + G(M-Mc)/R1 + GMc(M-Mc)/R1

Essentially, I have the total energy of the core + energy of the envelope + gravitational potential energy due to attraction between the core and envelope is the total energy of the star. I used the virial theorem, since we neglect radiation pressure/nuclear production, to get E = Ω/2. It should be algebra from here, but I am getting an overly complicated answer and am wondering if I missed something.

It should be easy to solve the equation for R1, and everything else is known. Then divide by R0 and you are done. Well, cancel G and see if something else can be simplified.

There are two squares missing on the right side.

## 1. How does the radius of a star change during core contraction?

The radius of a star decreases during core contraction. This is due to the inward gravitational force becoming stronger as the core collapses, causing the outer layers of the star to be pulled inwards.

## 2. What is the cause of a star's core contraction?

A star's core contracts when it runs out of its main source of fuel, hydrogen. As the hydrogen fuel is used up, the core can no longer produce enough energy to counteract the force of gravity, leading to collapse.

## 3. What happens to a star's temperature during core contraction?

The temperature of a star increases during core contraction. This is because as the core collapses, the pressure and density increase, causing the temperature to rise. This increase in temperature can trigger nuclear fusion of heavier elements, which can temporarily halt the contraction process.

## 4. How long does core contraction last in a star?

The duration of core contraction varies depending on the mass of the star. Smaller stars, such as red dwarfs, can undergo core contraction for billions of years before stabilizing. Larger stars, such as red giants, can experience rapid core contraction over a shorter period of time before evolving into a white dwarf or exploding as a supernova.

## 5. Can core contraction cause a star to change its spectral type?

Yes, core contraction can cause a star to change its spectral type. As the core contracts, the outer layers of the star heat up, causing the star to emit higher energy radiation and appear bluer. This change in temperature can also affect the chemical composition of the star's atmosphere, altering its spectral type.

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