ZHS and EP theory

ZHS and EP theory

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

C. W. James, Columbus, Ohio, Feb 23rd 2012 2

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:

Radiation termEnergy/area as R-2

Energy transport to infinity

Nearfield TermEnergy/area as R-4

Energy decreases with distance

“Accelerating charged particles radiate”

Get rid of this(no Frank-Tamm VC)


Do some maths…

• Endpoints

• ZHS formula

• Endpoints -> ZHS: (the far-field approximation)

**

*


Toy experiments

• 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


What do we expect to see?

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

Vavilo

v-Che

renko

v

radiat

ion im

porta

nt

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)

1 m

1000

m


Low-frequency-limit

• 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

• Observer much closer to track start than track end• Endpoints accounts for this, ZHS can not

1 m

1 m


Why?

• 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 vs ZHAires).- Important for dense media?...


What about near the Cherenkov angle?

• 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)

1 m1000 m


Why do endpoints blow up?• 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.

14

What happens in the near-field in the Cherenkov

regime?OR:

When good techniques go bad

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


Toy experimental set-up

• Place the observer firmly in the Cherenkov regime 10 m

1 m

Cherenkov zone


Spectrum: n=2

• Now we see differences…


Time-domain (no band limit)

• 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)


Quick check: in vacuum

• We do not expect and Cherenkov shock• But we do expect two bremsstrahlung

shocks…

• I do not understand this ZHS behaviour


1cm from the vacuum track

• Large ZHS pulse… in a vacuum.

• This is not V-C radiation!• It is a numerical artefact OR a static term.


Summary from toy experiments

• 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


Main conclusion

• Neither endpoints nor ZHS get it completely right

ZHS is better(probably not correct)

Far-field Near-fieldN

ear θ

CFa

r fro

m θ

C

EP & ZHS agree(probably correct)

EP theory is better(probably correct)

ZHS crazyEP misses VC (main)(probably both crap)


Philosophical aside

• What about smooth particle motion?

• Radiation is emitted constantly

• 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


What does the ZHS formula produce• 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


Is the divergence physical?

• 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)

ZHS and EP theory

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