Question

Radius-Mass Relation for p-p Burning Stellar Homologous ZAMS Stars For this problem, we will focus on one simple overall scaling for stellar radii R and masses M, which is easily obtained from the equations of stellar structure: \[ \frac{dr}{dm(r)} = \frac{4Tr^2p(r)}{dP(r)} \frac{Gm(r)}{dm(r)} \frac{4TTA}{dL(r)} = \frac{e(r)}{dm(r)} \frac{dT(r)}{3KRL(r)} \frac{dm(r)}{16acT^3r} \] (1) We will assume the stellar interior is supported solely by ideal gas pressure, which is appropriate for low-mass stars like our sun (graduate students will demonstrate why from the Eddington model on the last problem on this set), and with energy transport determined by a Kramers opacity law. We will further assume we can represent the volumetric nuclear energy generation rate as a power-law, \( \epsilon(r) = \epsilon_0 \rho^a T^b \), as we demonstrated in class. For this problem, we will consider stars burning through the p-p cycle of hydrogen fusion, where \( a = 5 \). Using these relations, and the luminosity-mass relation for low-mass stars derived in class \( L = 17.5M^{5.5} R^{0.5} \), demonstrate that homologous ZAMS stars have a radius-mass relation \( R \propto M \), where \( a \) and \( b \) are two scaling exponents which you will derive. Setting the scaling coefficient to solar values, you therefore have: \[ R = R_0 \left( \frac{M}{M_0} \right) \] (6)

Name: 2radius mass relation for p p burning stellar homologous zams stars for this problem we will focus on one simple overall scaling for stellar radii r and masses mwhich is easily obtained from 82024
Uploaded: 2023-07-25T13:16:04-08:00
Duration: 51 s
Channel: Numerade
Description: 2radius mass relation for p p burning stellar homologous zams stars for this problem we will focus on one simple overall scaling for stellar radii r and masses mwhich is easily obtained from 82024

          Radius-Mass Relation for p-p Burning Stellar Homologous ZAMS Stars

For this problem, we will focus on one simple overall scaling for stellar radii R and masses M, which is easily obtained from the equations of stellar structure:

\[ \frac{dr}{dm(r)} = \frac{4Tr^2p(r)}{dP(r)} \frac{Gm(r)}{dm(r)} \frac{4TTA}{dL(r)} = \frac{e(r)}{dm(r)} \frac{dT(r)}{3KRL(r)} \frac{dm(r)}{16acT^3r} \] (1)

We will assume the stellar interior is supported solely by ideal gas pressure, which is appropriate for low-mass stars like our sun (graduate students will demonstrate why from the Eddington model on the last problem on this set), and with energy transport determined by a Kramers opacity law. We will further assume we can represent the volumetric nuclear energy generation rate as a power-law, \( \epsilon(r) = \epsilon_0 \rho^a T^b \), as we demonstrated in class. For this problem, we will consider stars burning through the p-p cycle of hydrogen fusion, where \( a = 5 \).

Using these relations, and the luminosity-mass relation for low-mass stars derived in class \( L = 17.5M^{5.5} R^{0.5} \), demonstrate that homologous ZAMS stars have a radius-mass relation \( R \propto M \), where \( a \) and \( b \) are two scaling exponents which you will derive. Setting the scaling coefficient to solar values, you therefore have:

\[ R = R_0 \left( \frac{M}{M_0} \right) \] (6)

2radius mass relation for p p burning stellar homologous zams stars for this problem we will focus on one simple overall scaling for stellar radii r and masses mwhich is easily obtained from 82024

Added by Lisa N.

Question

Please give Ace some feedback