1 / 24

# ZHS and EP theory - PowerPoint PPT Presentation

ZHS and EP theory. C. W. James, Columbus, Ohio, Feb 23 rd , 2012. Step 1: Liénard-Weichert Potentials. Begin with Maxwell’s equations Add a single ( monopolar ) particle as a source Allow for finite light propagation speed Use Lorentz gauge. Step 2:. Apply to get:. Nearfield Term

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' ZHS and EP theory' - dena

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### ZHS and EP theory

C. W. James, Columbus, Ohio, Feb 23rd, 2012

Step 1: Liénard-Weichert Potentials

• Begin with Maxwell’s equations

• Add a single (monopolar) particle as a source

• Allow for finite light propagation speed

• Use Lorentz gauge

• Apply to get:

Nearfield Term

Energy/area as R-4

Energy decreases with distance

Energy/area as R-2

Energy transport to infinity

Get rid of this

(no Frank-Tamm VC)

Do some maths…

*

*

*

• Endpoints

• ZHS formula

• Endpoints -> ZHS:

(the far-field approximation)

• Take a straight particle track:

• Place an observer in x-z plane

• Calculate emission via…

• Endpoints

• ZHS (single track)

• ZHS (very many sub-tracks)

*

*

R

• Afanasiev, Kartavenko, Stepanovsky J Phys D, 32 (1999)

Bremsstrahlung from endpoints dominates

Bremsstrahlung from endpoints dominates

Far-field, far from theta_C

• Endpoints, ZHS agree perfectly.

• No ZHS track sub-division needed (1m source at 1 km unresolved)

1000 m

• Endpoints reduce to:

• ZHS low-phase limit:

• Tends towards a constant term at low frequencies

• Tends towards zero at low frequencies

Difference in the near-field

1 m

• Observer much closer to track start than track end

• Endpoints accounts for this, ZHS can not

• ZHS formula:

• Accounts for distance difference in phase, but not magnitude

• true no matter how tracks are subdivided

• Endpoints:

• Distance affects both magnitude and phase

• Clearly, an observer in the nearfield should see a monopolar component to the pulse

• [total net change in potential]

• Important for:

• Lunar Cherenkov? No! (very far field)

• Important for air-showers? Perhaps (REAS3 vsZHAires).

• Important for dense media?...

• Endpoint formulation:

• In ZHS:

• Result can be arbitrarily large (it blows up)

• Result is always finite (more sensible)

Behaviour near the Cherenkov angle

• Endpoints produce a larger contribution (can be arbitrarily large)

1000 m

• Endpoints allow:

• Infinitely small acceleration zone

• Infinitely small source particle

• Infinitely small detector

• [time-domain only] constant refractive index

• Result: potentially infinite field

• This should not be unexpected!

• Very common to see infinities in the literature

• This is why textbooks always derive the total radiated power and not the field strengths.

• This is small consolation.

• OR:

C. W. James, Columbus, Ohio, Feb 23rd 2012

• Place the observer firmly in the Cherenkov regime

10 m

1 m

Cherenkov zone

• Now we see differences…

• Time-domain output (ZHS vs EP) (n=2):

• Large contribution from ZHS NOT in endpoints!

• Could this be a ‘true’ Vavilov-Cherenkov emission? (or a numerical artefact?)

(note different y-axis scales)

• We do not expect and Cherenkov shock

• But we do expect two bremsstrahlung shocks…

• I do not understand this ZHS behaviour

• Large ZHS pulse… in a vacuum.

• This is not V-C radiation!

• It is a numerical artefact OR a static term.

• Theoretical expectation:

• EP theory models only bremsstrahlung

• Handles near-field

• Breaks down near theta_C

• ZHS models only bremsstrahlung + far-field approx

• Breaks down in near-field

• Handles theta_C

• What we see:

• EP theory matches expectation

• ZHS: some strange results…

• Produces phantom Vavilov-Cherenkov-like pulse

• Somehow misses bremsstrahlung

• Neither endpoints nor ZHS get it completely right

Far-field

Near-field

EP & ZHS agree

(probably correct)

EP theory is better

(probably correct)

Far from θC

ZHS crazy

EP misses VC (main)

(probably both crap)

ZHS is better

(probably not correct)

Near θC

• What about smooth particle motion?

• Limit (description -> perfection) [inf points]:

• Endpoints have contributions equal-and-opposite sides of the Cherenkov angle

• Divergences are expected to cancel

• Hence tendency towards ZHS treatment in REAS3

• ZHS formula approximates:

• This approximation can not be made near the Cherenkov angle

• Same approximation as Tamm (1939)

• Shown to exclude Frank-Tamm Cherenkov

• And yet…

• ZHS formula produces something sensible.

• Endpoints do not.

• We do not know what ZHS produces at the Cherenkov angle

• If:

• n is constant

• The acceleration event is truly instantaneous

• The particle and detector are both infinitely small

• Then yes!

• Divergence/magnification at the Cherenkov angle does NOT necessarily mean Vavilov-Cherenkov radiation!

• Q: Why do we often see total radiated power calculated, but not the fields?

• A: Because this can hide nasty divergences (integrate away this divergence over finite spatial angles)