# Deriving the Claim: \sin^2\theta/r Conservation in Pulsar Gamma-Ray Emission

• rbwang1225
In summary, the conversation discusses the mechanism of gamma-ray emission from pulsars and how the quantity sin2Θ/r is conserved along dipolar magnetic field lines. The conversation also explores how the gradient of this quantity being perpendicular to the field lines shows that it is constant along those lines. This type of derivation can be found in texts on vector analysis or electromagnetism.
rbwang1225
When I study the mechanism of gamma-ray emission from pulsars, I got a statement saying that the quantity $\sin^2\theta/r$ is conserved along any dipolar magnetic field line.

Does anybody know how to derive this claim?

The field of a magnetic dipole is B = (m/r3)(3r(r·z) - z) where r, z are unit vectors.

Let c = sin2Θ/r = (r2 - z2)/r3. Take the gradient of c and show that it is perpendicular to B. This shows that the lines c = const are the magnetic lines of force.

Bill_K said:
The field of a magnetic dipole is B = (m/r3)(3r(r·z) - z) where r, z are unit vectors.

Let c = sin2Θ/r = (r2 - z2)/r3. Take the gradient of c and show that it is perpendicular to B. This shows that the lines c = const are the magnetic lines of force.

Sorry, I do not understand why the gradient of c is perpendicular to B means that the lines c = const are the magnetic lines of force. And is c a constant?

Regards

rbwang1225 said:
Sorry, I do not understand why the gradient of c is perpendicular to B means that the lines c = const are the magnetic lines of force. And is c a constant?

Regards

If the gradient, i.e. the "change" of some quantity is perpendicular to some other field, it means that the quantity has no change in the direction of the field. It is therefore constant along the lines of that field.

Wow! I got it, but could you tell me what kind of text have this kind of derivations? calculus? or vector analysis?

Thanks!

rbwang1225 said:
Wow! I got it, but could you tell me what kind of text have this kind of derivations? calculus? or vector analysis?

Thanks!

It should be covered in any introduction to vector analysis. Texts on electromagnetism might also help.

## What is the significance of \sin^2\theta/r in pulsar gamma-ray emission?

The quantity \sin^2\theta/r is a measure of the angle at which the pulsar's magnetic field lines intersect with the observer's line of sight. This angle plays a crucial role in determining the intensity and polarization of gamma-ray emission from the pulsar.

## How is the conservation of \sin^2\theta/r related to pulsar gamma-ray emission?

The conservation of \sin^2\theta/r is a fundamental principle in the study of pulsar gamma-ray emission. It states that the ratio of the pulsar's magnetic field strength to its radius must remain constant in order to maintain a stable emission process. This principle helps explain the observed properties of pulsar gamma-ray emission and provides insight into the physical mechanisms at work.

## What evidence supports the conservation of \sin^2\theta/r in pulsar gamma-ray emission?

There have been numerous observations and studies that support the conservation of \sin^2\theta/r in pulsar gamma-ray emission. These include observations of the polarization and spectral properties of gamma-ray emission, as well as simulations and theoretical models that reproduce the observed trends. Additionally, the conservation of \sin^2\theta/r is consistent with other known properties of pulsars, such as their spin-down rates.

## Are there any exceptions to the conservation of \sin^2\theta/r in pulsar gamma-ray emission?

While the conservation of \sin^2\theta/r is a robust principle, there are some exceptions that have been observed in certain pulsars. These exceptions are thought to be due to the complex and dynamic nature of pulsar magnetospheres, which can lead to variations in the magnetic field geometry and emission processes. However, these exceptions are relatively rare and the conservation of \sin^2\theta/r remains a valid principle in the majority of cases.

## How does the conservation of \sin^2\theta/r relate to other conservation laws in physics?

The conservation of \sin^2\theta/r is a specific application of the more general concept of conservation laws in physics. These laws state that certain physical quantities, such as energy, momentum, and charge, must remain constant in a closed system. The conservation of \sin^2\theta/r is a unique manifestation of this principle in the context of pulsar gamma-ray emission, but it shares the same underlying concept of conservation.

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