# Integrating dP/dr over the Sun

• CaptainEvil
In summary, the conversation discusses estimating the central pressure of the Sun by integrating the equation of hydrostatic equilibrium from the Sun's radius to its core. The estimated value, using a simplified density profile, is 1.4e6 atm. It is noted that this value is much lower than the true value, but is sufficient for the purposes of the question. The question is then raised about the consistency of this value with the original question.
CaptainEvil

## Homework Statement

Consider the Sun to be a sphere of hydrogen gas of uniform density equal to its
average density (1440 kg/m3).
a) Integrate the equation of hydrostatic equilibrium for dP/dr from the Sun’s radius to the
core to estimate the central pressure in Pa and in atmospheres. Assume the pressure at the
surface is zero. [Note: your estimate will fall far short of the true value that can be
derived using a more realistic density profile, about 250 billion atmospheres, but that’s ok
for our purposes]

## Homework Equations

dP/dr = -Gmp/r^2 (p = density)

## The Attempt at a Solution

I replaced m with 4/3pir^3p which yields
dP/dr = -4piGrp^2/3
and integrated from R to 0 yielding 2piGp^2R^2/3
and filling in the values and Radius of the sun, I get 1.4e6 atm of pressure.
Does this make sense? (Is it consistent with what the question is asking)

I would say that your attempt at a solution is a good start to estimating the central pressure of the Sun. However, there are a few things to consider when integrating dP/dr over the Sun.

First, the equation of hydrostatic equilibrium is usually written as dP/dr = -Gm(r)p(r)/r^2, where m(r) is the mass enclosed within a radius r and p(r) is the density at that radius. Since the Sun is a sphere of uniform density, your equation for dP/dr is correct, but it would be more accurate to write it as dP/dr = -4piGr^2p.

Second, when integrating over the entire Sun, you should use the total mass of the Sun, not just the mass enclosed within a certain radius. This means that you should use the mass of the entire Sun (M = (4/3)piR^3p) in your integration, not just (4/3)piR^3p.

Finally, your estimate of 1.4e6 atm is significantly lower than the true value of 250 billion atmospheres. This is because your calculation assumes a uniform density throughout the Sun, which is not the case in reality. The density of the Sun increases significantly towards the core, leading to much higher central pressures.

In conclusion, your attempt at a solution is a good start, but to get a more accurate estimate of the central pressure of the Sun, you would need to consider the varying density profile and use the total mass of the Sun in your integration.

## 1. What is the purpose of integrating dP/dr over the Sun?

The purpose of integrating dP/dr over the Sun is to calculate the total pressure gradient force acting on the Sun's surface. This is important in understanding the dynamics and behavior of the Sun's atmosphere.

## 2. How is dP/dr calculated and integrated over the Sun?

dP/dr is calculated by taking the derivative of the pressure with respect to the radial distance from the center of the Sun. This derivative is then integrated over the entire surface of the Sun using mathematical techniques such as Riemann sums or numerical integration methods.

## 3. What factors influence the dP/dr over the Sun?

The dP/dr over the Sun is influenced by the temperature, density, and composition of the Sun's atmosphere. It is also affected by the Sun's magnetic field and any internal processes such as convection or nuclear reactions.

## 4. What does the value of dP/dr tell us about the Sun?

The value of dP/dr can tell us about the overall structure and behavior of the Sun's atmosphere. A high value of dP/dr indicates a strong pressure gradient force, which can lead to phenomena such as solar flares or coronal mass ejections. On the other hand, a low value of dP/dr may indicate a more stable and quiescent atmosphere.

## 5. How is integrating dP/dr over the Sun useful in studying solar weather and space weather?

Integrating dP/dr over the Sun is useful in studying solar weather and space weather because it provides information about the forces acting on the Sun's atmosphere. Understanding these forces is crucial in predicting and monitoring events such as solar flares, coronal mass ejections, and other space weather phenomena that can impact Earth and our technological systems.

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