- #1

rbwang1225

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Does anybody know how to derive this claim?

Thanks in advance.

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- Thread starter rbwang1225
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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.

- #1

rbwang1225

- 118

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Does anybody know how to derive this claim?

Thanks in advance.

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- #2

Bill_K

Science Advisor

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Let c = sin

- #3

rbwang1225

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Bill_K said:B= (m/r^{3})(3r(r·z) -z) wherer,zare unit vectors.

Let c = sin^{2}Θ/r = (r^{2}- z^{2})/r^{3}. Take the gradient of c and show that it is perpendicular toB. 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

Regards

- #4

Polyrhythmic

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rbwang1225 said:Sorry, I do not understand why the gradient of c is perpendicular toBmeans 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.

- #5

rbwang1225

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Thanks!

- #6

Polyrhythmic

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rbwang1225 said:

Thanks!

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

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.

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.

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.

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.

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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