astro-ph/9501030 10 Jan 95 a stars radio sup ernovae as well as sup ernova remnants there app ears to b e cosmic rays we prop ose an explanation for the observed protonelectron ratio in galactic I Introduction as regards the implicationsvarious of imp ortant these caveats and concepts outlinecosmic for imp ortant ray stars next data and steps available stellardata as from evolution well from other as a checks exp eriments Akenotively b on a world cosmic data ray set sp c ectrum Flys and Eye chemical and d comp osition against against further airshower the often stars that have a stellar wind prior to the explosion of the toroidalmay eld put with additional reduced momentumdwarfs into massive angular stellar red winds giant stars fromnetic and eld the Wolf strengths pressure Rayet which stars gradient inner may convection b ecome zone directly of observableical an in sho cks upp massive er in white mainplasma a sequence we stellar star derive can wind the lead prop erties to of high mag energetic particles accelerated in spher can ing argument as wellfor as the the magnetic derivation eld of strengths the maximum particle energy that an statistics due to sho ck structures such a stellar wind of what happ ens when the star explo des and a sho ckwave travels down ergies up torays near of the nucleicle of acceleration Helium which andstars we heavier suggest elements can fromserved explain GeV through most in nonthermal of particle radio emission in the en novae observed OB cosmic stars and Wolf Rayet Max Planck Institut f urRadioastronomie BonnPeter Germany L Biermann THEIR SUPERNOVA BLASTWAVES AND PRESUPERNOVA WINDS go through Sho cks in stellar winds can accelerate particles energetic particles are ob critical underlying physics b e Most sup ernovae are explosions of massive stars see Sup ernova explosions into predecessor stellar winds can lead to parti reached of Alfvenic COSMIC RAY CONTRIBUTION the following steps at GeV energies Wolf of these principle From Rayet Machnumber 9 stellar winds Optical and Xray data have b een GeV as well as electrons ab ove ab out GeV We that stars and sho ckheating a comparison leads We suggest that a dynamo working in the and We check to make the case for This limit gives supp ort to the wind driv radiosup ernovae to electron b ecomes transp ort of the momentum loss the mo del injection eg imp ortant for a discussion radioluminosities of various co ecients Finally we summarize Using a p ostulate Owocki to Such magnetic elds prediction quantita derive With interpreted in We use the eg a this a lower turbulent Wheeler concept Obser Thus limit for as

PL BIERMANN

vationally there is also evidence for sho ckwaves in stellar winds prior

to the explosion such as suggested by nonthermal radio emission from

OB and Wolf Rayet stars as well by the nonthermal radio emission

from the nova GK Per Seaquist et al An interpretation of

this nonthermal radio emission is an imp ortant test for any theory of

particle acceleration in sho cks in stellar winds

In this chapter we will review recent work on the acceleration pro

cess in sho ck waves in stellar winds and argue that a large part of the

observed cosmic rays can b e attributed to sup ernova sho cks in stellar

winds

A Stellar winds

Stellar winds are observed in many cases low mass stars such as

the Sun as well as high mass stars such as OB stars For the latter

the wind driving can b e explained as an eect of radiation pressure

Lucy Solomon Castor et al Pauldrach et al

Owocki For Wolf Rayet stars the momentum in the wind is

generally b elieved to b e to o large to b e explainable as due to radiation

driving in the limit of single scattering and so as one p ossible solution

multiple scattering mo dels have b een devised Magnetic elds have

b een argued to contribute to the driving through the fast magnetic

rotator concept Cassinelli this idea however has b een

criticized on the basis that the corresp onding large angular momentum

loss would lead to a severe selflimitation of the pro cess Nerney Suess

and that therefore the pro cess could not b e general There is a

mo died magnetic rotator mo del for which the Alfvenicsurface is close

to the stellar surface and hence the angular momentum loss is strongly

reduced Biermann Cassinelli In this case the lifetime is not

limited by angular momentum loss but an initial driving of the wind

is required and the pressure gradient of the tangential magnetic eld

is argued to provide an amplication of the momentum of the wind

B Cosmic Rays

After the discovery of cosmic rays by Hess and Kohlhorster

Baade Zwicky already prop osed that sup ernova explo

sions pro duce cosmic rays Alfven argued early for a lo cal origin

in our Galaxy which is conrmed by the age determinations of the

cosmic rays GarciaMunoz et al Fermi prop osed

the basic concept of acceleration still b eing used Shklovskii

and Ginzburg made a convincing case for particle acceleration

in sup ernova remnants Ginzburg already emphasized the inter

esting role of novae and indeed the nova GK Per is a test case for the

evolution of sho cks in winds and their particle acceleration Hayakawa

prop osed that stellar evolution gives rise to an enhancement of heavy elements and p ointed out the imp ortance of spallation in the in

STELLAR WINDS SNe and CRs

terstellar medium and again the enrichment found in the cosmic ray

contribution from sup ernova explosions into winds is indeed b elieved

to day to b e due to just this enrichment And nally Co cconi al

ready argued convincingly that the most energetic cosmic ray particles

are from an extragalactic origin the GRO observations Sreekumar et

al of a neighboring galaxy the Small Magellanic Cloud provided

the last and a very strong argument that indeed the cosmic rays in the

lower energy range are not universal and thus have to b e galactic An

early seminal form of some of the ideas expressed in the following can

b e found in Peters A brief historical review is given by

Ginzburg The form of Fermis pro cess used to day was discov

ered nearly simultaneously by Axford et al Bell a b

Blandford Ostriker and Krymskii In this picture the

main particle energy gain is from rep eated scattering o magnetic ir

regularities on the two sides of a sho ck front Since those two sides

can b e considered as a continuously contracting system relative to each

other particles that remain in the system gain energy

There are many imp ortant reviews and b o oks on cosmic ray phys

ics We just mention the classical b o ok by Hayakawa the new

b o ok by Berezinsky et al and the reviews by Drury

Blandford Eichler and Jones Ellison

Today there are only some well accepted arguments ab out the ori

gins of cosmic rays a The cosmic rays b elow ab out GeV are

b elieved to b e predominantly due to the explosion of stars into the

normal interstellar medium Lagage Cesarsky b The cosmic

rays from near GeV up to the knee at GeV are likely pre

dominantly due to explosions of massive stars into their former stellar

wind Volk Biermann c While this latter claim is not undis

puted there has b een certainly no agreement yet on the origin of the

cosmic rays of higher particle energy

Direct observation of cosmic rays either from goundbased instru

mentation from satellites or from ballo ons has b een the driving input

for nearly all considerations in cosmic ray work These data demon

strate that i the overall sp ectrum of cosmic rays is a p owerlaw up to an

energy commonly referred to as the knee near GeV to continue

with a steep er p owerlaw to the ankle near GeV with a slight

turnup b eyond and an apparent cuto near GeV ii the sp ectra

are ab out E for Hydrogen at mo derate energy and slightly atter

for Helium and heavier elements while the electron sp ectrum is con

sistent with the Hydrogen sp ectrum at low energy and then changes

over to ab out E iii the chemical comp osition at low energy is

crudely similar to that of the interstellar medium with Hydrogen and

Helium underabundant relative to Silicon All such prop erties require

explanation

We will assume in the following that the correction from the ob

PL BIERMANN

served cosmic ray sp ectrum to the source sp ectrum is a change in sp ec

tral slop e by exactly for relativistic particles see Biermann a

a b c so that we are lo oking at source sp ectra of approximately

E b elow the knee and of approximately E ab ove the knee This

can b e argued on the basis of a Kolmogorov sp ectrum of the irregular

ities of the interstellar magnetic eld from plasma simulations from

analogies with in situ measurements of the solar wind and other ob

servations of the interstellar medium see eg Biermann a It is

clear however that a Kolmogorov sp ectrum of interstellar turbulence

is insucient to explain in a straightforward manner eg the abun

dance ratios in cosmic rays see eg GarciaMunoz et al and

Biermann c Such prop erties of cosmic rays may require a deep er

understanding of the interstellar medium than we currently have How

ever we note that the secondary to primary ratio of spallation pro ducts

in cosmic rays do es not simply yield a sp ectrum of interstellar turbu

lence in a medium which varies on time scales equivalent to those of

cosmic ray transp ort and which has known inhomogeneities of density

contrasts of many p owers of ten

C Sho cks in winds

There are various kinds of evidence for the presence of sho ck struc

tures in stellar winds from in situ observations in the solar wind to

optical line proles to Xray emission from massive stars Lucy

Owocki Rybicki Owocki et al Owocki

MacFarlane Cassinelli Usov Melrose Feldmeier

Furthermore there are sho cks to b e exp ected in the colliding wind

zones b etween the binary star comp onents of two massive stars Eich

ler Usov we do not consider such cases Here we wish to

concentrate on particle acceleration and the ensuing nonthermal radio

emission from sho ck zones in the winds of single stars

Nonthermal radio emission is usually attributed to the synchrotron

emission of energetic electrons gyrating in magnetic elds OB and

Wolf Rayet stars are b oth stars with strong p owerful winds they are

usually thermal radio emitters from freefree emission and in many

cases they are also nonthermal radio emitters Similarly nonthermal

radio emission is detected from some novae in particular from the nova

GK Per Thus any theory to explain particle acceleration in spherical

sho cks has to account for the prop erties of such nonthermal radio emis

sion if as a result one nds an explanation for a comp onent of cosmic

rays it provides supp ort for the coherence of a theory to account for

the origin of cosmic rays

For this nonthermal radio emission we can determine its luminos

ity sp ectrum time dep endence spatial distribution and what turns

out to b e esp ecially useful its p olarization In sup ernova remnants and in the nova GK Per we can even determine the spatial arrangement of

STELLAR WINDS SNe and CRs

p olarization This allows us to derive the top ology of the magnetic

eld For young sup ernova remnants this spatial arrangement is al

ways predominantly radial while the compression of an arbitrarily ar

ranged interstellar magnetic eld would naturally lead to a tangential

conguration Also for the wind of the nova GK Per we exp ect an

unp erturb ed tangential magnetic eld and a strengthening of such a

eld in a sho ck wave the observations also in this case show a radial

eld Seaquist et al The observed top ology of the magnetic

eld is thus at right angles to that exp ected on very simple grounds

This implies a very general reason and we would would like to suggest

that this is b est understo o d as the consequence of a rapid convective

radial motion in the sho cked region of the plasma If this notion is

the correct interpretation of these data then the diusive transp ort of

energetic particles in the sho cked region is p ossibly dominated by this

rapid convective motion and its prop erties then inuence the sp ectrum

of the energetic particles in the sho ck

This leads to the basic thesis underlying all the arguments made

here We prop ose that a principle of the smal lest dominant scale ei

ther in real space or in velocity space Biermann a allows us to

determine the relevant transp ort co ecients which describ e the overall

transp ort of particles in the sho cked region b oth parallel and trans

verse to the sho ck direction and to derive drift energy gains

D Explosions Interstellar medium versus Winds

One might ask what the essential dierence actually is b etween

explosions into a stellar wind and an explosion into the interstellar

medium There are three imp ortant dierences i one is that a stellar

wind has a density gradient asymptotically of density r r with

radius r leading to p ersistent high sho ck velocities ii the second is

the p ossibility that stellar winds have much stronger magnetic elds

than the interstellar medium and iii that the winds of massive stars

are enriched in heavy elements in the last stages of their evolution thus

strongly biasing any energetic particle p opulation

As a consequence the rst phase of an explosion when the ejected

mass M is in free expansion until an approximately equal amount of

ej

mass is snowplowed together has a quite dierent characteristic radius

R For an interstellar medium of ion density n this radius is

e i

M

ej

n R ISM p c

e

i

M

Usually this radius is exceeded even for low densities of the inter

stellar medium so that the phase of expansion with constant energy

is relevant for any discussion the Sedov phase This means that the

sho ck sp eed U is steadily decreasing with radius as U r

PL BIERMANN

In contrast the corresp onding radius for the expansion into a wind

is given by

M V

ej W

R w ind p c

e

M

M

Here V is the wind velocity in units of c and M is the

W

wind mass loss in units of M p er year This radius is obviously

reduced for the explosive expansion into a red giant wind with a wind

sp eed of only kmsec by a factor of to only a few parsec ev

erything else b eing equal However even then this radius will usually

not b e reached or at least not b e surpassed by a considerable factor

As a result the sho ck velocity of an explosion into the interstellar

medium is typically slowing down steadily while in striking contrast

the explosion into a wind can b e exp ected to b e in an approximately

free expansion phase and so at nearly constant high velocity

Another question one might ask is whether an explosion into

a wind do es not naturally lead to an explosion into the interstellar

medium when the edge of the wind bubble is reached From evolution

ary calculations of massive stars it is clear that there is considerable

mass loss b efore the nal explosion and all this mass forms together

with snowplowed interstellar medium material a thick shell around the

wind zone Hence for those stars which have a large amount of mass

loss through a wind the sup ernova sho ck has some diculty p ene

trating this shell around the wind bubble without considerable energy

loss One dierence b etween wind sup ernovae and interstellar medium

sup ernovae is then that in the rst case the explosion dissipates a signif

icant fraction of its energy in the wind bubble shell while in the second

case the explosion can break through and form a remnant in adiabatic

expansion Wheeler shows that the dividing line is likely to b e

near a zeroage main sequence mass of ab out solar masses we discuss

this p oint b elow section VI in the context of the energy requirements

of the galactic cosmic rays attributed to the two source p opulations

E Outline

The review is based on earlier work summarizes it and expands

up on it Biermann a pap er CR I Biermann Cassinelli

pap er CR I I Biermann Strom pap er CR I I I Stanev et al

pap er CR IV Rachen Biermann pap er UHE CR I Rachen et al

pap er UHE CR I I Nath Biermann a b Biermann

Gaisser Stanev with earlier reviews in Biermann b a

b c with however an entirely dierent emphasis

Much of the new material describ ed here is based on discussions

the author had with other participants at the meetings in Raleigh

NC September organized by D Ellison S Reynolds in Tuc

son AZ Octob er organized by CP Sonett MS Giampapa

STELLAR WINDS SNe and CRs

JR Jokipii in Budap est Hungary March organized by Zs

Nemeth and E Sormorjai in Vulcano Italy May organized by

F Giovannelli and G Manno cchi in Nandaihe China August

organized by ZG Deng XY Xia and G Bornerin St Petersburg

Russia September organized by DA Varshalovich A Bykov

and others and in Sto ckholm Sweden September organized

by L Bergstrom P Carlson PO Hulth and H Snellman Not all

rep orts written for these meetings are mentioned

In the following we will describ e in section I I the basic concept for

particle acceleration in spherical sho cks in section III we will derive

the prop erties of the energetic particle p opulation in section IV we will

discuss the implications for stellar winds in section V we will discuss

the consequences for energetic electrons in section VI for energetic

nuclei and summarize in section VI I

I I The sho ck region

A A basic instability

The normal asymptotic conguration for an embedded magnetic

eld in a stellar wind has b een derived by Parker and gives a

magnetic eld which is tangential decreases with radius r as r and

with colatitude as sin

Consider a spherical sho ck wave in such a wind We will use the ap

proximation throughout that the sho ck wave is spherically symmetric

and that any asymmetry is introduced by the orientation and latitude

dep endence of the magnetic eld only This is a strong simplication

but is necessary to keep the issue clear which we discuss here the phys

ics of sho ck waves which are at least lo cally spherical and which run

through a stellar wind

Then a spherical sho ck wave centered on the star in its symme

try compresses the magnetic eld this b eing tangential by the full

compression factor of for a strong sho ck in an adiabatic gas of index

Particles can b e injected into an acceleration pro cess and give a

large prop ortion of the overall pressure and energy density see eg

Ellison et al Jones Ellison Then we have the case of a

cosmic ray mo died sho ck wave such as treated by Zank et al

In the unp erturb ed state we have here a conguration where the mag

netic eld is p erp endicular to the sho ck direction ie parallel to the

sho ck surface Zank et al demonstrated that for this case the congu

ration is neutrally stable with an increasing instability as so on as the

sho ck b ecomes more oblique relative to the underlying magnetic eld

leading to maximum instability for the case of a parallel sho ck congu

ration In the oblique sho ck conguration as many as three instabilities

PL BIERMANN

may op erate see their Table In the case which we consider here

this means that the slightest p erturbation of the sho ck surface relative

to the underlying magnetic eld is leading to an instability Such an

instability then increases the deformation of the surface which in turn

strengthens the instability Ultimately the sho ck surface is maximally

deformed and changes its shap e continuously since the conguration

is stuck in an unstable mo de Thus strong turbulence is exp ected in

the sho ck region including downstream Ratkiewicz et al

This means then for instance that at the tip of an outward bulge

of the sho ck surface the bulge can move outward with resp ect to the

time averaged sho ck frame until a column density is encountered equal

to the column density b ehind the sho ck then the bulge driven by the

instability has to slow down in its lo cal motion At the sides of the

bulge energetic particles readily can leak into the upstream region go

ing through the sho ck just once in a lo cally highly oblique conguration

ie magnetic eld versus sho ck normal b efore the sho ck overruns

them again This is equivalent to saying that the lo cal acceleration

eciency is strongly reduced b ecause here we envisage an energetic

particle to sp end a fair amount of time on either side of the sho ck

gaining an appreciable amount of energy from drifts b efore going back

through the sho ck

Such a picture then leads to a concept where the sho ck surface

is jumping around an average lo cation a sphere in our case The up

stream length scale of this jumping is a length corresp onding to the

same column density as the average downstream region which corre

sp onds to all the matter snowplowed in the expansion of the spherical

sho ck It is imp ortant to note that the length scales asso ciated with

the instability are hydrodynamic and therefore we b elieve to b e justi

ed to use hydrodynamic scales b elow we also adopt b elow the gross

simplication of the maximum density jump of valid for a normal gas

with adiabatic co ecient of here ignoring any cosmic ray mo dica

tion see eg Duy et al In this concept the acceleration is a

combination of a the drifts the particles exp erience in the upstream or

downstream regions since the averaged magnetic elds are of course

still p erp endicular to the sho ck direction and b the energy gain from

the Lorentz transformation each time a particle go es through the sho ck

At the same time particles also lose energy from adiabatic expansion

since in the expansion of a spherical sho ck the lo cal length scales always

increase

We have to caution that we use here results from cosmic ray trans

p ort theory in an environment for which this theory was not made ie

where the magnetized ionized plasma is dominantly turbulent

B Radiop olarization

How can we test such a concept with observations The radio ob

STELLAR WINDS SNe and CRs

servations of young sup ernova remnants and the nova GK Per can b e

a guide here Radio p olarization observations of sup ernova remnants

clearly indicate what the typical lo cal structure of these sho cked plas

mas is Obviously the normal expansion of a sup ernova remnant is not

into a stellar wind but into the interstellar medium However the typ

ical magnetic eld is highly oblique on average in a random eld and

so the essential issue remains the same as in the case of a sho ck into

a wind The observational evidence Milne Downs Thompson

Reynolds Gilmore Milne Dickel et al has

b een summarized by Dickel et al in the statement that all shell

type sup ernova remnants less than years old show dominant ra

dial structure in their magnetic elds near their b oundaries There are

several p ossible ways to explain this we concentrate here on the idea

that this p olarization pattern is due to rapid convective motion which

induces lo cally strong shear

These examples are for sup ernova explosions into the interstellar

medium there is also an observation demonstrating the same eect for

an explosion into a wind Seaquist et al nd for the spatially

resolved shell of the nova GK Per a radially oriented magnetic eld in

the shell while the overall dep endence of the magnetic eld on radial

distance r is deduced to b e r just as exp ected for the tightly wound

up magnetic eld in a wind The interpretation given is that the shell

is the higher density material b ehind a sho ck wave caused by the nova

explosion in and now travelling through a wind Seaquist et al

note the similarity to young sup ernova remnants

The imp ortant conclusion for us here is that there app ear to b e

strong radial dierential motions in p erp endicular sho cks which pro

vide the p ossibility that particles get convected parallel to the sho ck

direction The instability describ ed earlier Zank et al may b e

the physical reason for this rapid convective motion we would like to

suggest We assume this to b e a diusive pro cess and note that others

have also p ointed out that this may b e a key to sho ck acceleration eg

Falle

It is imp ortant to emphasize that older sup ernova remnants do

not show a clear pattern as describ ed ab ove Already the data of the

radio knot motions in Cas A Tus clearly show a rather chaotic

b ehaviour it app ears that the radio knots move erratically with a sp eed

of the order of the sho ck sp eed itself This is actually imp ortant it

turns out we will use this erratic motion b elow to limit the drift eect

along the sho ck sphere

C Tycho versus Cygnus Lo op

One imp ortant consequence of the concept introduced here is that

for spherical sho cks that do not accelerate a new particle p opulation to high energy the overall shell thickness should b e just that due to the

PL BIERMANN

snowplow for a spherical sho ck in a homogeneous medium this is r

in the case of a strong sho ck Obviously if co oling b ecomes imp ortant

then the thickness is even less For a sho ck that do es accelerate a new

energetic particle p opulation in a sho ck in the interstellar medium the

instability describ ed ab ove leads to an estimate of the shell thickness

of r again for a strong sho ck and referred to the outer edge see

b elow

The Cygnus Lo op as well as Tycho app ear to show indeed such

prop erties the Cygnus lo op do es not require a new particle p opulation

for its radio emission the squeezed interstellar medium is well sucient

and the shell is thin Raymond p erscomm Green at the

same time the sup ernova remnant Tycho Dickel et al do es

require a newly accelerated particle p opulation to explain the radio

observations and its shell thickness over most of its circumference is

close to prediction ie fairly thick Also here we note that such an

agreement do es not prove our concept to b e correct but it do es show

consistency We will discuss this p oint at some more length b elow

I I I The sp ectrum

A Derivation

The imp ortant conclusion for us here is that observations and the

oretical arguments suggest the existence of strong radial dierential

motions in those p erp endicular sho cks which are mediated by cosmic

rays this in turn suggests that particles get convected parallel to the

sho ck direction We emphasize that convective motion at a given scale

entails that particle diusion is indep endent of energy We assume this

convective turbulence with asso ciated particle transp ort to b e a dif

fusive pro cess for which we have to derive a natural velocity and a

natural length scale which can b e combined to yield a diusion co ef

cient A classical prescription is the metho d of Prandtl In

Prandtls argument an analogy to kinetic gas theory is used to derive

a diusion co ecient from a natural scale and a natural velocity of the

system Despite many weaknesses of this generalization Prandtls the

ory has held up remarkably well in many areas of physics far b eyond

the original intent In a leap of faith we will use a similar prescription

here

In order to generalize we introduce the notion of the smal lest dom

inant scale This can b e a scale in length or in velocity may refer to an

anisotropic transp ort and thus b e dierent in orthogonal directions

This principle do es not say that nature lets convective transp ort com

p ete on the basis of dierent scales b ecause then the longest scale

would b e the fastest transp ort and would win it rather implies that

STELLAR WINDS SNe and CRs

nature chooses for the eective convective transp ort the smallest dom

inant scales The idea that the smallest dominant scale may eg b e

related to the distance to a wall in a uid current has already b een

used for a long time see section I I I x in Prandtl We use the as

sumption that the smallest dominant scale is the relevant scale several

times in the course of this work in order to derive diusion co ecients

and other scalings In some cases when a smallest dominant scale is

energy dep endent there may b e a switch from one physical scale to

another say to an energyindep endent scale Such a switch denes

critical particle energies

Consider the structure of a layer sho cked by a Sup ernova explosion

into a stellar wind in the case that the adiabatic index of the gas is

and the sho ck is strong Then there is an inherent length scale in

the system namely the thickness of the sho cked layer in the spherical

case for a sho ck velocity much larger than the wind sp eed and in the

strong sho ck limit r This is the thickness of the matter snowplowed

all the way from the star to the current lo cation of the sho ckfront in

the simplied picture of constant density throughout the shell There

is also a natural velocity scale namely the velocity dierence of the

ow with resp ect to the two sides of the sho ck Both are the smallest

dominant scale in velocity and in length

Our basic conjecture argument based on observational evidence

as wel l as theoretical arguments is then that the convective random

walk of energetic particles p erp endicular to the unp erturb ed magnetic

eld can b e describ ed by a diusive pro cess with a downstream diusion

co ecient which is given by the thickness of the sho cked layer and

r r

the velocity dierence across the sho ck and is indep endent of energy

U

r U U

r r

U

This is in apparent contradiction to the normal argument that for

relativistic particles c E c r where E is the mean

r r g

free path for resonant scattering of a particle of energy E and Larmor

radius r We do not invoke resonant scattering but fast convective

g

turbulence as the dominant pro cess We do not ignore resonant scat

tering but use Jokipiis argument on p ermissible scattering co

ecients in oblique geometries see just b elow to demonstrate that we

are within b ounds here

The upstream diusion co ecient can b e derived in a similar way

but with a larger scale We make here the second critical step argu

ment namely that the upstream length scale is just U U times

larger and so is r This is the relevant scale for the same column den

sity on b oth sides of the average sho ck lo cation and can b e argued on

the basis of what limits the instability see ab ove This also is the

same ratio as the mass density and the ratio of the gyroradii of the

PL BIERMANN

same particle energy Since the magnetic eld is lower by a factor of

U U upstream that means that the upstream gyroradius of the max

imum energy particle that could b e contained in the sho cked layer is

also r Hence the natural scale is just r And so the upstream diusion

co ecient is

r U U

r r

It immediately follows that

U

r r r r

r U r U U

For these diusion co ecients it also follows that the residence

times Drury on b oth sides of the sho ck are equal and are

U r

r r r r

U c U c c U

Here it has to b e stated that the residence time is normally derived

from a diusion argument which is obviously not valid here b ecause it

requires that the scattering length is much smaller than the dominant

scale while we basically identify those two scales We have not proven

but surmise that the concept of the residence time could b e rederived

from a timeaveraging of the probability that a given particle is still on

the same side of the jumping and wobbling sho ck surface after some

time t It is an assumption based on dimensional arguments that the

result would b e the same

Adiabatic losses then cannot limit the energy reached by any par

ticle since they run directly with the acceleration time b oth b eing

indep endent of energy and so the limiting size of the sho cked layer

limits the energy that can b e reached to that where the gyroradius just

equals the thickness of the sho cked layer provided the particles can

reach this energy We assume here that the average of the magnetic

eld hB i is not changed very much by all this convective motion This

then leads to a maximum energy of

U

Z er B Z er B E

max

U

where Z e is the particle charge and B is the magnetic eld strength

on the two sides of the sho ck This means that the energy reached

corresp onds to the maximum gyroradius the system will allow on b oth

sides of the sho ck It also says that we push the diusive picture right

up its limit where on the downstream side the diusive scale b ecomes

equal to the mean free path and the gyroradius of the most energetic particles

STELLAR WINDS SNe and CRs

Jokipii has derived a general condition for p ossible values of

the diusion co ecient Its value has to b e larger than the gyroradius

multiplied by the sho ck sp eed This condition is fullled here for the

maximum energy particles only by a factor of U U since

here the sho ck sp eed and the radial scale of the system give b oth the

largest gyroradius as well as the diusion co ecient This also counters

part of the criticism of Volkin Ellison et al the other part of

his criticism is dealt with in section of pap er CR I

Observations can now give information of p ossible values for the

diusion co ecient as well Smith et al estimate the cosmic ray

diusion co ecient at sho cks in the LMC and nd severe upp er limits

However rst of all they use the assumption that the underlying un

p erturb ed magnetic eld is parallel to the sho ck normal second they

measure an instantaneous situation whereas I have argued ab ove that

strong and fast convective motion dominates the sho ck region The dif

fusion co ecient derived ab ove is an eective temp oral and geometric

average over this fast convection Hence there is no contradiction

There is an imp ortant consequence of this picture for the diu

sion laterally From the residence timescale and the velocity dierence

across the sho ck we nd a distance which can b e traversed in this time

of

r U

U U

c U

Since the convective turbulence in the radial direction also induces

motion in the other two directions with maximum velocity dierences

of again U U this distance is also the the typical lateral length

scale We noted ab ove that the observed motions of radio knots in

the sup ernova remnant Cas A supp ort such an argument Tus

From this scale and again the residence time we can construct an upp er

limit to the diusion co ecient in lateral directions of

U U

r c

max

U c

which is for strong sho cks equal to

U

r c

max

c

Again in the spirit of the idea that the smallest dominant scale

determines the eective transp ort this then will b egin to dominate as

so on as the diusion co ecient reaches this maximum at a critical

energy As long as the diusion co ecient is smaller it will dominate

particle transp ort in and the upp er limit derived here is irrelevant

When the diusion co ecient reaches and passes this maximum then the particle in its drift will no longer see an increased curvature due to

PL BIERMANN

the convective turbulence b ecause of averaging and the part here we

have to account for b oth losses and gains by drifts of drift acceleration

due to increased curvature is eliminated This then reduces the energy

gain and the sp ectrum b ecomes steep er from that energy on The

critical particle energy thus implied will b e identied b elow with the

particle energy at the knee of the observed cosmic ray sp ectrum

A Drifts

Consider particles which are either upstream of the sho ck or down

stream as long as the gyro center is upstream we will consider the par

ticle to b e there and similarly downstream

In general the energy gain of the particles will b e governed primar

ily by their adiabatic motion in the electric and magnetic elds The

expression for the energy gain is then Northrop eq for

an isotropic angular distribution

dE U B pw l nB

Z eV

d

dt c c t

i

where the rst term arises from the drifts and the second from the

induced electric eld This equation is valid in any co ordinate frame

We explicitly work in the sho ck frame and the separate the two terms

ab ove and consider the drift term rst The second term is accounted

for further b elow

The drift velocity in a normal stellar wind is given ab ove The

drift can b e understo o d as arising from the asymmetric comp onent

of the diusion tensor the r comp onent The natural scales there are

the gyroradius and the sp eed of light and so we note that for Forman

et al

r c

g r

the exact limiting form derived from ensemble averaging we obtain the

drift velocity by taking the prop er covariant divergence Jokipii et al

this is not simply spherical co ordinates the r derivative of

r

The general drift velocity is given by see eg Jokipii

B E

cur l V c

d

Z e B

The drift velocity is thus

c r r V

g d

where r is now taken to b e p ositive This drift velocity is just that

g

due to the gradient as well as the curvature and in fact b oth eects contribute here equally

STELLAR WINDS SNe and CRs

It must b e remembered that there is a lot of convective turbulence

which increases the curvature The characteristic scale of the turbu

lence is r for strong sho cks and thus the curvature is r maximum

Here we have to consider the balance of drift energy gains and losses

We call the motion of a convective element that moves in the upstream

direction upow and for a downstream direction of motion we call it

downow Then the direction of an upow convective element is not in

general just radial and so we have to take the average of an assumed

omnidirectional distribution of upow directions weighting it with the

probability of that particular direction and obtain of the maxi

mum curvature same averaging as done in eq of Drury for

the averaging of the momentum gain It is plausible that for upow

directions we have to take the maximum curvature since the velocity

dierence to the surroundings is maximum Downow however the

velocity dierence to the surroundings is only of that upow and

so we take for the downow convection a curvature of of the maxi

mum multiplied also by The dierence of gains and losses is then

the net gain thus giving a factor of of the max

imum curvature argument This is equivalent to taking half the

maximum as average for the net energy gain due to drifts We obtain

then for the curvature a factor of r which is twice the curvature with

out any turbulence this increases the curvature term by a factor of two

thus changing its contribution from to in the numerical factor

in the expression ab ove Hence the total drift velocity combining now

again the curvature and gradient terms is thus

U

V c r r

d g

U

now written for arbitrary sho ck strength It is easily veried that the

factor in front is unity for strong sho cks where U U

The energy gain asso ciated with such a drift is given by the pro duct

of the drift velocity the residence time and the electric eld Upstream

this energy gain is given by

U U

E E f

d

c U

where

U

f

d

U

Thus f for strong sho cks The corresp onding expression down

d

stream is

U U

E E f

d

c U

PL BIERMANN

giving a total energy gain of

U U U

E E f

d

c U U

The drift energy gain averages over the magnetic eld strength dur

ing the gyromotion We emphasize that this energy gain is indep endent

of this average magnetic eld so that even variations of the magnetic

eld strength due to convective motions do not change this energy gain

It can easily b e shown that this treatment can b e extended to sub

relativistic energies and provides p owerlaws in momentum as a result

just as in standard case see b elow and Drury

It is of interest to note here that the net distance travelled ie

drifted by the particle eg upstream is given by

U E

f l

d ?

Z eB U

which is the gyroradius itself for U U corresp onding to a

strong sho ck This then says that we are at a gyroradius limit for

the drift distance Just as in isotropic turbulence the gyroradius is

a lower limit to the mean free path for particle scattering parallel to

the magnetic eld in a turbulent plasma suggesting that it may b e

useful to think of the plasma also as maximally turbulent p erp endicular

b oth to the ow and to the magnetic eld We emphasize that during

this drift the particle makes many gyromotions It is also imp ortant

to note that the magnetic eld structure in the sho cked region as

discussed ab ove on the basis of observations will contain lo cal regions

of opp osite magnetic eld and so the drift itself will b e erratic and

b e the sum of many single element drift movements What we have

derived is the average net energy gain due to drifts with the drift

distance corresp onding to the average magnetic eld strength

A The energy gain of particles

Sho ck acceleration in its standard form just uses the Lorentz trans

formation for an energetic particle at a velocity v much larger than the

sho ck velocity to compute the energy gain as the particle go es b etween

scattering in a weakly turbulent magnetic eld from downstream to

upstream and back In practice we will consider the case when v is

close to c

We assume the sho ck to b e subrelativistic and so the phase space

distribution of the particles to b e nearly isotropic Then downstream

see Drury for the exact derivation the particles have a nite

chance to escap e A detailed discussion yields a p owerlaw for the dis

tribution p f p where f p is the particle distribution function of

momentum p in phase space with the p owerlaw index for strong

sho cks for a gas of adiabatic index ie p f p p

STELLAR WINDS SNe and CRs

However this assumes that there are no other energy losses or en

ergy gains during a cycle of a particle going back and forth b etween

upstream and downstream In a conguration where there is a com

p onent of the magnetic eld p erp endicular to the sho ck normal there

can b e energy gains by drifts parallel to the electric eld seen by a

particle moving with the sho ck system and also losses due to adiabatic

expansion in the expansion of a curved sho ck Since drift acceleration

is a rate the resulting particle energy gain is prop ortional to the time

sp ent on either side of the sho ck Furthermore there can b e sp ectral

changes due to the fact that particles were injected at a dierent rate

in the past when some particle under consideration now was injected

again this is likely to happ en in a spherical sho ck

Let us consider then one full cycle of a particle remaining near the

sho ck and cycling back and forth from upstream to downstream and

back The energy gain just due to the Lorentz transformations in one

cycle can then b e written as

U U E

E c U

LT

Adding the energy gain due to drifts we obtain

U U E

x

E c U

where

U U

x

U U

which is for a strong sho ck when U U

It is easy to show that the additional energy gain attens the par

ticle sp ectrum by

U

U U x

A Expansion and injection history

Consider how long it takes a particle to reach a certain energy

U dt U

r r

xE g f gf

dE U c c U

Here we have used that U U Since we have

r r r r

r U t

this leads to

PL BIERMANN

dt U dE

r r

t E U U x r U

and so to a dep endence of

E

tE t

o

E

o

with

U

r r

U U x r U

which is a constant indep endent of r and t

Particles that were injected some time ago were injected at a dier

b

ent rate say prop ortional to r This then leads to a correction factor

for the abundance of

E

b

E

o

d

However in a ddimensional space particles have r more space

available to them than when they were injected and so we have another

correction factor which is

E

d

E

o

The combined eect is a sp ectral change by

U

r r

d b

U U x r U

Thus we have a density correction factor which dep ends on the

particle energy and so changes the sp ectrum

This expression can b e compared with a limiting expansion de

rived by Drury eq who also allowed for a velocity eld

Drury generalized earlier work on spherical sho cks by Krymskii

Petukhov and Prishchep Ptuskin Drurys expres

sion agrees with the more generally derived expression given here for

x The comparison with Drurys work claries that for r the

inherent time dep endence drops out except obviously for the highest

energy particles discussed further b elow the same comparison shows

that the statistics of the pro cess are prop erly taken into account in our

r r1

is in the wind case and simplied treatment We note that

r U

1

in the ISM case b oth for strong sho cks and thus still small com

pared to unity A fortiori the comparison shows that in this limit of

r r1

the derivation by Drury using the prop erly derived classical small

r U

1

cosmic ray transp ort theory and our heuristic derivation b oth agree

STELLAR WINDS SNe and CRs

This agreement do es not provide pro of of our microscopic picture but

it is an imp ortant check

d

If the expansion is linear as it is the case here then the r term

also describ es the adiabatic losses in their eect on the sp ectrum due

to the general expansion of the sho ck layer and thus accounts for the

second term in eq ab ove Hence the total sp ectral dierence as

compared with the planeparallel case is given by

U U

r r

f b d g

U U U x x r U

Here we use the following sign convention For this expression p os

itive the sp ectral index of the particle distribution is steep er than

without this correction this then takes the minus sign in eqs

prop erly into account

This expression constitutes the basic result of this section

For a wind we have b and d and so b d For

a Sedovtype expansion into a homogeneous medium the additional

dierential adiabatic losses steep en the sp ectrum further pap er CR

I I I The total sp ectral change is then for U U given by so

that the sp ectrum obtained is

Sp ectrum source E

This is what we wanted to derive Generalizing to transrelativis

tic energies we note that the sp ectrum is a p owerlaw in relativistic

momentum p as can easily b e seen from the detailed derivation

Generalizing now for arbitrary wind sp eed and arbitrary sho ck

strength we obtain a reduced thickness of the sho cked layer This

reduction of the thickness of the layer for a nite wind velocity is due

to the fact that the material which is snowplowed together is not all gas

b etween zero radius and the current radius r but b etween zero radius

and r V V U since the gas keeps moving while the sho ck

W W

moves out towards r For the sequence of V U and

W

we thus obtain particle sp ectral index dierences in addition to the in

dex of of corresp onding to synchrotron emission

sp ectral index of an electron p opulation with the same sp ectrum of

using the convention that ux density S

The work of Owocki et al suggests that typical sho cks in

winds have a velocity in the wind frame similar to the wind velocity

in the observers frame itself which implies that in the simplied pic

ture here only sp ectral indices for the synchrotron emission b etween

and are relevant with an extreme range of sp ectral in

dices up to for strong sho cks Obviously for weaker sho cks

with U U the sp ectrum can b e steep er eg for U U

we obtain an optically thin sp ectral index for the synchrotron emission

PL BIERMANN

of for V U and for V U This then is the

W W

prediction for the sp ectral index of the nonthermal radio emission from

massive stars like OB stars and Wolf Rayet stars as well as for other

sho cks in winds like around novae

We can estimate the uncertainty only with diculty since the ar

gument is a limit the limit of a strong sho ck for instance and also in

all other steps of the argument we have used the conceptual limit of

the scales involved However we can estimate one uncertainty which

arises from the nite wind sp eed of Wolf Rayet stars or similar stars

with strong winds which explo de as sup ernovae These wind sp eeds

can go up to several thousand kmsec while the sup ernova sho ck is

variously estimated to kmsec to twice that much As a limiting

argument we use that the ratio of the wind sp eed to the sup ernova

sho ck sp eed is this gives a steep ening of the derived sp ectral

index of the particle distribution by This uncertainty also may

corresp ond to curvature of the sp ectrum since there is a timeevolution

as the sho ck progresses out through the stellar wind As more energy

of the sho ck is dissipated and more mass of the stellar wind snow

plowed the sho ck slows down then those particles already accelerated

keep their atter sp ectrum see eq of Drury while those

particles freshly injected and accelerated will have a steep er sp ectrum

Thus in the range V U we obtain a sp ectral index in the

W

range We note that the evolved stars which are most

common have a slow wind and so the distribution of stars through this

adopted range of wind velocities is likely to b e biased towards the small

numbers Therefore we ascrib e to the sp ectral index derived here an

uncertainty of which describ es b oth the uncertainty in

an assumed p owerlaw and the p ossible curvature After correcting for

leakage from the Galaxy the sp ectrum is



Sp ectrum earth E

very close to what is observed near earth at particle energies b elow the

knee We plan to discuss the error estimates in a separate contribution

It is of interest to note that an injection sp ectrum of E can

also lead via ppcollisions in a synchrotron dominated regime to a sp ec

trum of pairsecondaries of E which translates in the synchrotron

sp ectrum to a p owerlaw ux density sp ectrum and a p ow

erlaw photon number sp ectrum very close to that observed by GRO

for the Crab pulsar Schonfelder seminar in Bonn If the cur

vature of a sho ck is not exactly spherical obviously some dierences

to these particular numerical values will o ccur Such a sp eculative in

terpretation would place the origin of the pulses in p erio dically excited

sho cks travelling down a p erp endicular magnetic eld conguration in a pulsar wind as considered here

STELLAR WINDS SNe and CRs

Such an injection sp ectrum of of relativistic particles in strong

and fast sho cks propagating through a stellar wind leads to an unam

biguous radio synchrotron emission sp ectrum of in ux density

S compare eg the nonthermal radio emission of OB stars WR stars

novae esp ecially GK Per Reynolds Chevalier radio su

p ernovae and sup ernova remnants The observed variety of sp ectral

indices of the radio emission of sup ernova remnants is discussed in some

detail in pap er CR I I I

B The maximum energy of particles

The maximum energy particles can reach is given ab ove and de

p ends linearly on the magnetic eld Thus we require estimates for the

magnetic eld in the stellar winds of Wolf Rayet stars and other mas

sive stars that explo de as Sup ernovae like red and blue sup ergiants

Comparing at rst the corresp onding estimates that Volk Biermann

used we note that the energies implied here are larger by ap

proximately cU for the same given magnetic eld strength since their

expression for the maximum energy that particles could reach contains

an additional factor of approximately U c as compared with our

eq This mo died limit of Volk Biermann to the particle energy

which can b e reached corresp onds well to the limit valid for sup ernova

explosions into the interstellar medium see pap er CR I I I

Cassinelli Maheswaran Cassinelli have ar

gued that Wolf Rayet stars have very much larger magnetic elds than

Volk Biermann used in order to drive their winds The magnetic

elds given by Cassinelli and coworkers are of order a few thousand

Gauss on the surface of the star We introduce the conjecture here

discussed in more detail in pap er CR II and b elow that the Alfven

radius of the stellar wind is close to the stellar surface itself Then it

follows that the pro duct B r has approximately the same value on the

surface as in the wind and is of order B cm Gauss where B

is the strength of the magnetic eld at cm radial distance in units

of Gauss From this number we infer a maximum energy of particles

of

E protons B GeV

max

and

E iron B GeV

max

This suggests that the highest energy particles from the accelera

tion pro cess discussed here are mostly iron or other heavy nuclei The

chemical comp osition is exp ected to change abruptly to mostly pro

tons again when the extragalactic comp onent takes over pap er UHE

CR I somewhere near GeV The dep endence of the maximum

PL BIERMANN

particle energy on the magnetic prop erties of the stellar winds implies

a smearing due to the distribution of magnetic eld strengths from all

the dierent stars which contribute and hence leads to a steep ening

of the sp ectrum p ossibly well b efore the cuto If this mechanism

provides the largest particles energies then obviously other contribu

tions are not excluded by pulsars neutron star binaries or even from

a hypothetical termination of the galactic wind

C The knee in the Cosmic Ray sp ectrum

We wish to discuss here the b end in the sp ectrum of Cosmic Rays

at the knee near GeV

Let us consider the structure of the wind through which the sup er

nova sho ck is running The maximum energy a particle can reach is

prop ortional to sin since the space available for the gyromotion from

a particular latitude is limited in the direction of the p ole by the axis

of symmetry Hence the maximum energy attainable is lowest near

the p oles Then consider the p ole region itself where the radial de

p endence of the magnetic eld is r and the magnetic eld is mostly

radial We can make two arguments here Either we put the upstream

diusive scale c U equal to r c in the strong sho ck limit or

r r

we can put acceleration time and ow time equal to each other Both

arguments lead to the same result We use here the Bohm limit in

cE Z eB r since we have a sho ck the diusion co ecient

r r

conguration near the p ole where the direction of propagation of the

sho ck is parallel to the magnetic eld often referred to as a paral

lel sho ck conguration This then leads to a maximum energy for the

particles of

U

E Z eB r r

c

which is prop ortional to r near the p ole where the magnetic eld

is parallel to the direction of sho ck propagation the corresp onding

U

1

r Putting this equal to the gyroradius gyroradius is then given by

c

of particles that are accelerated further out at some colatitude where

the magnetic eld is nearly p erp endicular to the direction of sho ck

propagation gives the limit where the latitudedep endent acceleration

breaks down This then gives the critical angle as

U

sin

cr it

c

The angular range of we refer to as the polar cap b elow

cr it

We note that this angle is usually much larger than the angle where

B B at a given radial distance r If as observed in the solar wind

r

the equatorial sheet of the magnetic eld structure oscillates around

the geometric symmetry plane then this critical angle may either b e

STELLAR WINDS SNe and CRs

eectively larger or the argument may fail altogether The energy at

the lo cation of the critical angle as dened ab ove is then given by

U

E Z eB r r

k nee

c

We identify this energy with the knee feature in the Cosmic Ray

sp ectrum since all latitudes outside the p olar cap contribute the same

sp ectrum up to this energy from this energy to higher particle energies

a smaller part of the hemisphere contributes and also the energy gain

is reduced as argued b elow this reduction of the energy gain in each

cycle of acceleration happ ens at the same critical energy This is valid

in the region where the magnetic eld is nearly p erp endicular to the

sho ck and thus this knee energy is indep endent of radius

All this immediately implies that the chemical comp osition at the

knee changes so that the gyroradius of the particles at the sp ectral

break is the same implying that the dierent nuclei break o in order

of their charge Z considered as particles of a certain energy and not

as energy p er nucleon In a sp ectrum in energy p er particle this

introduces a considerable smearing

In the p olar cap the acceleration is a continuous mix b etween the

regime where the diusion co ecient is determined by the thickness of

the shell and the regime where it is dominated by turbulence parallel to

the magnetic eld this latter regime is rather small in angular extent

Thus r U might b e quite a bit smaller than Hence the p olar

r r

cap will have a sp ectrum which is determined by a range of

U

r r

r U U

as well as by a rather reduced role for the extra energy gain due to

drifts Thus the sp ectrum is harder in the p olar cap region b ecause

we are close to the standard parallel sho ck conguration for which the

particle sp ectrum is well approximated by E On the other hand

the p olar cap is small relative to with ab out U U and only a

sp ectrum much atter than E like eg indeed E will make it

p ossible for the p olar cap to contribute appreciably near the knee en

ergy b ecause then the sp ectral ux near the knee is increased relative

to GeV by E m c which approximately comp ensates for its

k nee p

small area The uncertainty in the sp ectrum of the p olar cap comp o

nent particles is not determinable at present should the sp ectrum b e

signicantly steep er than E then this comp onent almost certainly

is altogether insignicant Therefore we do not ascrib e any uncertainty

to this sp ectral index it has to remain an assumption derived from the

limiting argument During an episo de with drift towards the p oles a

larger part of the sphere can contribute for larger energy particles and so there is an additional tendency to atten the sp ectrum of the p olar

PL BIERMANN

cap contribution The combination of the p olar cap with the rest of the

stellar hemisphere might lead to a situation where up to say Z TeV

the entire hemisphere excluding the p olar cap dominates while from

Z TeV up to the knee the p olar cap b egins to contribute apprecia

bly Near the knee energy the p olar caps might thus contribute equally

to the rest of the steradians Because of spatial limitations most of

the hemisphere has to dominate again ab ove the knee although with

a fraction of the hemisphere that decreases with particle energy This

introduces a weak progressive steep ening of the sp ectrum with energy

which we will discuss elsewhere The sup erp osition of such sp ectra for

dierent chemical elements including the p olar cap contribution has

b een tested successfully see pap er CR IV and is briey summarized

b elow in section VI The results of these checks suggest that the p olar

cap may b e a reason for the attening of the cosmic ray sp ectrum as

one approaches the knee feature

We note that we are using a limiting argument to derive the sp ec

trum b elow the knee and again use a limiting argument see b elow

for the sp ectrum ab ove the knee Close to the knee such an argument

breaks down on either side and so a softening of the knee feature is

to b e exp ected On top of such a softened knee feature the p olar cap

gives an additional comp onent

The expression for the particle energy at the knee also implies by

the observed relative sharpness of the break of the sp ectrum that the

actual values of the combination B r r U must b e very nearly the same

for all sup ernovae that contribute appreciably in this energy range

however it is dicult to put a numerical value to this argument see

pap er CR IV for the systematics that go into any parameter estimate

from tting the airshower data Please note that B r r is evaluated

in the Parker regime and so is related to the surface magnetic eld

by B r r B r V where the values with index s refer to the

s s W

s

surface of the star and V is the wind velocity Thus in our picture

W

the expression

s

U B r

s

s

V

W

is approximately a universal constant for all stars that explo de as su

p ernova after a Wolf Rayet phase It may also hold for all massive stars

of lower mass that explo de as sup ernovae

This then implies that we may have identied a functional rela

tionship for the mechanical energy of explo ding stars connecting the

magnetic eld the angular momentum and the ejection energy Such

a relationship could b e fortuitous since all massive stars b ecome very

similar to each other near the end of their evolution But it could also

b e an indication for an underlying physical cause Related ideas have b een expressed and discussed by Kardashev BisnovatyiKogan

STELLAR WINDS SNe and CRs

LeBlanc Wilson Ostriker Gunn Amnuel et

al BisnovatyiKogan et al and Kundt with

BisnovatyiKogan the closest to the argument b elow All this

leads to the following interesting suggestion see pap er CR I

The source of the mechanical energy observable in sup ernova explo

sions may then b e the gravitational energy of a core accretion disk at a

scale determined by angular momentum and mediated by the magnetic

eld We leave a discussion how this argument may lead to a surface

magnetic eld strength to elsewhere

The distribution in the knee particle energy from the dierent stel

lar prop erties and explosive energies clearly leads to a rounding of the

eective average knee shap e which softens the p owerlaws on the b oth

sides of the knee see Biermann d

D The latitude distribution of the particles

Consider the derivation of the sp ectrum b eyond the knee Since the

maximum energy particles can attain is a strong function of colatitude

the sp ectrum b eyond the knee requires a discussion of the latitude

distribution which we have to derive rst The latitude distribution is

established by the drift of particles which builds up a gradient which in

turn leads to diusion down the gradient Hence it is clear that drifts

towards the equator lead to higher particle densities near the equator

and drifts towards the p oles lead to higher particle densities there

Thus the equilibrium latitude distribution is given by the balancing of

the diusion and the drift

The diusion tensor comp onent can b e derived similar to our

heuristic derivation of the radial diusion term again by using the

r r

smallest dominant scales The characteristic velocity of particles in

is given by the erratic part of the drifting corresp onding to spatial ele

ments of dierent magnetic eld direction This is on average the value

of the drift velocity j V j p ossibly mo died by the lo cally increased

d

values of the magnetic eld strength The characteristic distance is the

distance to the symmetry axis r sin this is the smallest dominant

scale as so on as the thickness of the sho cked layer is larger than the

distance to the symmetry axis ie sin U U Thus we can write

in this approximation argument

j V j r

d

Here is the cosine of the colatitude on the sphere we consider for

the sho ck in the wind Interestingly this can also b e written in the

form

r c

g

PL BIERMANN

where r is taken as p ositive we also note that c is the

g

maximum drift sp eed at a given latitude valid for the lo cal maximum

particle energy This suggests that the latitude diusion might b e use

fully thought of as diusion with a length scale of the gyroradius and

the particle sp eed to within the angle dep endent factor which just

cancels out the latitude dep endence of the magnetic eld strength in

the denominator of the gyroradius

We assume then for the colatitude dep endence a p owerlaw

a

and rst match the latitude dep endence of the diusion term

and the drift and then use the numerical co ecients to determine the

exp onent in this law The diusion term and the drift term have the

same colatitude dep endence since the double derivative and the internal

a

factor of lead to a for the diusive term while

the drift term is just the simple derivative giving the same expression

For the condition then is a a The diusive

term is always p ositive while the drift term is negative for ZB

s

negative This means for p ositive particles and a magnetic eld directed

inwards the drift is towards the p ole In that case then the exp onent

a is either zero or a Since the drift itself clearly pro duces

a gradient the case with a is of no interest here It follows

that for p ositive particles and an inwardly directed magnetic eld the

latitude distribution is strongly biased towards the p oles emphasizing

in its integral the lower energies and thus making the overall sp ectrum

steep er b eyond the knee energy such a conguration may inuence the

p olar cap structure in terms of magnetic elds and energetic particles

The radial drift in this case is directed outwards which means that

particles drift ahead of the sho ck by a small amount only to b e caught

up again by the diusive region ahead of the sho ck For the magnetic

eld directed outwards and p ositive particles the radial drift is inwards

taking particles out of the system at an slightly increased rate and thus

steep ening the overall sp ectrum by a small amount

When the magnetic eld is directed outwards and the particles have

a p ositive charge the drift is towards the equator with then a p ositive

gradient with again in the limit

We note that this exp onent is reduced in the case that the

erratic part of the drift is increased over the steady net drift comp onent

One imp ortant consequence exists for the resulting radio emission

When the density of energetic particles is largest near the equator

where the magnetic eld is strongest then we have the case of maximum

radio emission In the other extreme case that the density of energetic

particles is maximal near the p oles where the magnetic eld is minimal

then the radio emission is minimal ie unobservable usually All this

is relevant of course only in the simplied picture of a homogeneous

structure of the stellar wind If there were a rapid cycling of p olarities

eg much faster than on the Sun then the p erio d may interfere with

STELLAR WINDS SNe and CRs

this picture the consequences have not b een worked out yet

E The sp ectrum b eyond the knee

This means that from E the energy gain of all particles in

k nee

the entire colatitude range that is aected by diusion has to b e

considered together

In our mo del for the diusion in we have used the drift velocity

and the distance to the symmetry axis as natural scales in velocity

and in length When the drift reaches the maximum derived earlier

eq then the latitude drift changes character This happ ens at a

critical energy which is reached at

max

which translates into see eq

U

E E E

cr it max k nee

c

We emphasize that two dierent basically geometric arguments

lead to the same critical energy E since we have derived the same

k nee

critical energy from arguments ab out the p olar cap However it is from

this argument here that the sp ectrum b eyond the knee follows

Ab ove this critical particle energy then the particles average in

their path over the small curvature scales and p erceive dominantly the

full curvature of the system Since at full curvature r there is no

direction ambiguity the same argument applied as ab ove indicates that

we have a net balance of for the curvature term reducing it

from its value at U U of to ie in eq the numer

ical value of the rst bracket go es from to thus reducing the

value of x from to This then leads to an overall sp ectrum

of E b efore taking leakage into account We emphasize that this

result is based on putting all the parameters which go into it at their

most simple or most extreme limit Again using the correction implied

by a nite wind sp eed of V U gives a mo died sp ectrum of

W

E Therefore we asso ciate an error range of with the sp ec

tral index derived ab ove We note in addition that in our mo del where

the sup ernova explosions into the interstellar medium and the sup er

nova explosions into stellar winds b oth contribute to mo derate particle

energies these dierences in the sources may imply dierences in the

propagation which can only b e quantied after a thorough investiga

tion of the cosmic ray propagation in the Galaxy Thus nally the

sp ectrum is



E

PL BIERMANN

with leakage accounted for This is what we wanted to derive As noted

ab ove the use of limiting arguments to derive the sp ectrum on either

side of the knee implies that the knee itself may b e quite soft and thus

curvature is to b e exp ected see Biermann d

F Assumptions and Systematic Uncertainties

The assumptions adopted are inspired by Prandtls mixing length

approach all use the key prop osition that the smallest dominant

scale either in geometric length or in velocity space gives the natural

transp ort co ecient In this sense the assumptions are derived from a

basic principle which we p ostulate

Our basic argument based on observational evidence as wel l

as theoretical reasoning is that for a cosmic ray mediated sho ck the

convective random walk of energetic particles p erp endicular to the un

perturbed magnetic eld can b e describ ed by a diusive pro cess with a

downstream diusion co ecient which is given by the thickness

r r

of the sho cked layer and the velocity dierence across the sho ck and

is indep endent of energy

The upstream diusion co ecient can b e derived in a similar way

but with a larger scale based on the same column density as in the

downstream layer This leads to the second critical argument

namely that the upstream length scale is just U U times larger

It must b e remembered that the large intensity of convective turbu

lence increases the curvature The characteristic scale of the turbulence

is r for strong sho cks again as an example in the case of the wind

SN and thus the curvature is r maximum Half the maximum of the

curvature allows for the net balance of gains and losses for the energy

gain due to drifts argument and so we obtain then for the cur

vature r which is twice the curvature without any turbulence this

increases the curvature term for the sp ectral range b elow the knee

The diusion tensor comp onent can b e derived similar to our

heuristic derivation of the radial diusion term again by using the

r r

smallest dominant scales The characteristic velocity of particles in

is given by the erratic part of the drifting corresp onding to spatial

elements of dierent magnetic eld direction This is on average the

value of the drift velocity j V j p ossibly mo died by the lo cally in

d

creased values of the magnetic eld strength The characteristic length

is the distance to the symmetry axis r sin argument this is the

smallest dominant scale as so on as the thickness of the sho cked layer

is larger than the distance to the symmetry axis ie sin U U

Rapid convection also gives a comp eting diusion in the direction

indep endent of particle energy this will b egin to dominate as so on

as the energy dep endent diusion co ecient reaches this maximum

at a critical energy As long as the diusion co ecient is smaller

it will dominate particle transp ort in and the upp er limit derived

STELLAR WINDS SNe and CRs

here is irrelevant When the diusion co ecient reaches and passes

this maximum given by the fast convection then the particle in its

drift will no longer see an increased curvature due to the convective

turbulence due to averaging The part of drift acceleration due to

increased curvature is eliminated Again a detailed consideration of

gains and losses of the drift energy gains leads to the sp ectrum of

particles b eyond the knee The critical energy derived in this way is

the same as that derived from a phasespace argument near the p oles

All these arguments are inspired by Prandtls mixing length ap

proach all use the key prop osition that the smallest dominant scale

either in geometric length or in velocity space gives the diusive trans

p ort discussed We assume this to b e true even for the anisotropic

transp ort parallel and p erp endicular to the sho ck

In addition we i use the simplied notion of a purely spherical

sho ck ii ignore the mo dications of the sho ck introduced by the cos

mic rays themselves except in the conceptual derivation of the initial

argument where the cosmic rays are critical for the instability iii use

a test particle approach iv have not checked with a full D calcula

tion on a sup ercomputer system that our conceptualization of the lo cal

physics actually yields what we argue and v also have not checked

with such a calculation that our concept of the smal lest dominant scale

is as general as we use it We note however that this concept is al

ready contained in early descriptions of Prandtls mixing length scale

argument

We have to emphasize very strongly that these uncertainties mean

that the sp ectral indices derived for the p owerlaw section of the various

comp onents of the cosmic rays corresp ond to a limiting argument If

things were really not as simple and they are likely to b e much more

complicated then the sp ectrum derived and any comparison with data

has to b e taken with considerable caution

G Summary of the predictions

The prop osal is that three sites of origin account for the cosmic rays

observed i sup ernova explosions into the interstellar medium ISM

SN ii sup ernova explosions into the stellar wind of the predecessor

star windSN and iii radio galaxy hot sp ots Here the cosmic rays

attributed to sup ernovashocks in stellar winds windSN pro duce an

imp ortant contribution at all energies up to GeV

Particle energies go up to Z TeV for ISMSN and to Z

PeV with a b end at Z TeV for windSN Radiogalaxy hot sp ots go

up to near or slightly b eyond EeV at the source These numerical

values are estimates with uncertainties of surely larger than a factor of

since they derive from an estimated strength of the magnetic eld

and estimated values of the eective sho ck velocity see ab ove



The sp ectra are predicted to b e E for ISMSN pap er CR

PL BIERMANN

 

I I I and for windSN E and E b elow and

ab ove the knee resp ectively and E at injection for radiogalaxy hot

sp ots The p olar cap of the windSN contributes an E comp onent

allowing for leakage from the Galaxy which however contributes

signicantly only near and b elow the knee if at all The uncertainty

in the radiogalaxy sp ectra will b e discussed elsewhere These sp ectra

are for nuclei and are corrected for leakage from the galaxy Electron

sp ectra are discussed b elow

The chemical abundances are near normal for the injection from

ISMSN and are strongly enriched for the contributions from windSN

Helium and heavier elements are dominantly from windSN already at

GeV particle energies At the knee the sp ectrum b ends downwards at

a given rigidity and so the heavier elements b end downwards at higher

energy p er particle Thus b eyond the knee the heavy elements dominate

all the way to the switchover to the extragalactic comp onent which is

once again mostly Hydrogen and Helium corresp onding to what is ex

p ected to contribute from the interstellear medium of a radiogalaxy as

well as from any intergalactic contribution mixed in Biermann b

This continuous mix in the chemical comp osition at the knee already

renders the overall knee feature in a sp ectrum in energy p er particle

unavoidably quite smo oth a tendency which can only partially b e o

set by the p ossible p olar cap contribution since that comp onent also

is strongest at a given rigidity

We note that further uncertainties of the sp ectrum derive from

a the time evolution of any acceleration pro cess as the sho ck races

outward b the match b etween ISMSN and windSN c the mixing

of dierent stellar sources with p ossibly dierent magnetic prop erties

and d the dierences in propagation in any mo del which uses dierent

source p opulations These uncertainties translate into a distribution

of p owerlaw indices of the sp ectra to curvature of the sp ectra to a

smearing of the knee feature and to a smo othing of the cutos Obvi

ously this is in addition to the underlying uncertainty asso ciated with

the concept of the smal lest dominant scale itself

There is one imp ortant prediction for stars The particle energies

of cosmic rays in the windcomp onent go up to Z PeV which nec

essarily implies that those massive stars which have strong winds into

which they explo de as sup ernovae have appreciable magnetic elds at

that stage The magnetic eld is required to b e of order Gauss at a

distance of cm from the hydrostatic surface of the star if the mag

netic eld top ology is of a Parker type ie is mostly tangential and

dep ends on radius r as r right down to near the surface of the star

as argued in pap er CR I I and b elow then the surface magnetic elds

are still quite mo derate eg of order a few thousand Gauss for Wolf

Rayet stars We will explore some of the consequences in the following

STELLAR WINDS SNe and CRs

IV Winds of massive stars

In this section we discuss the winds of massive stars and the conse

quences of the prop osition that the magnetic elds may not b e totally

negligible ie may b e of the order of Gauss at cm radial distance

in the stellar wind

A Dynamo

First we prop ose that high magnetic elds can b e generated inside

massive stars so that later when these interiors get exp osed to b ecome

the surface of Wolf Rayet stars these high magnetic elds can help drive

the wind The dynamo mechanism is b elieved to act in the turbulent

zone in the interior of rotating massive stars and given a seed eld

such as may b e pro duced by a battery mechanism in a rotating star

increases the magnetic eld strength up to a maximum which can b e

estimated in dierent ways One p ossible limit is the dynamic pressure

of the convective motions The analogy with the interstellar medium

suggests the limit where the Coriolis force equals the magnetic stresses

Ruzmaikin et al Gilb ert Childress Gilb ert The

strength of the magnetic eld can then b e estimated from the condition

that the magnetic torque is limited by the Coriolis forces over the size

of the convective region This condition can b e written as

B R v f

c cc t

where is the rotation rate of the star is the average density of

c

the convection zone R is the radius of the convective core v is the

cc t

characteristic turbulent velocity and f is a correction factor of order

unity in order to allow for a structural variations ignored here and also

for b the fact that the dominant length scale is likely to b e smaller

than the radius of the convective region The turbulent velocity is

estimated from the condition that turbulent convection transp orts all

the luminosity L

L R v f

c

cc t

where f is also a correction factor of order unity to allow for structural

variations we use the average density and the radius of the entire

convective region

Mo dels of Langer p erscomm provide the input together

with data from the textb o ok of Cox Giuli

M

R R

M

M

M M

cc

M

PL BIERMANN

M

R R

cc

M

where R and M are the stellar radius and mass resp ectively This

leads to a magnetic eld of

M

Gauss B f f

r ot

M

where is the fraction of critical rotation at the surface assumed to

r ot

b e solid b o dy rotation The same argument for Wolf Rayet stars using

data of Langer with Helium mass fraction Y gives

M

W

B f f Gauss

r ot

M

Hence this readily pro duces magnetic elds dep endent on the ro

tation rate of the star of up to Gauss for O stars and ab out

times more for Wolf Rayet stars In b oth cases the induced magnetic

eld is nearly indep endent of stellar mass Critical rotation here means

that the convective core has to rotate at an angular velocity which

would corresp ond at the surface to critical rotation there but in fact

as we will see b elow it is not required that the actual surface rotates

this fast

In conclusion we nd that we rather easily generate magnetic elds

at levels deep inside the star b eyond the lo cal virial limit at the surface

of the star which gives of the order of a few times Gauss nearly in

dep endent of the rotation rate Maheswaran Cassinelli

Rotationally induced circulations may b e able to carry these magnetic

elds to the surface of the stars on a time scale much shorter than

the main sequence life time In this transp ort the magnetic eld is

weakened by ux conservation and so surface elds in the range

to Gauss are quite plausible and are b elow the lo cal virial theo

rem limit Such values are all we require in the following It is these

weaker magnetic eld strengths which we will use in the following so

that the uncertainty of what really limits the dynamo mechanism can

b e checked more from observation than theoretical faith These esti

mates are valid for massive stars and extend clearly to b elow the mass

range where we have Wolf Rayet stars as an imp ortant nal phase of

evolution We thus exp ect many of the arguments in the following to

hold generally for all massive single stars with extended winds whether

slow or fast whether red or blue sup ergiant preceding the sup ernova

explosion

There is a further consequence for the generation of magnetic elds

in white dwarfs The most massive stars that do not b ecome sup er

novae but white dwarfs are suciently massive to contain also con

vective cores which get exp osed when the white dwarf is formed Thus

STELLAR WINDS SNe and CRs

there ought to b e a correlation b etween massive white dwarfs and the

detection of strong magnetic elds this is consistent with the observa

tional data Lieb ert p erscomm Schmidt et al

An interesting additional observation is the high incidence of kilo

Gauss elds on Am stars Lanz Mathys three of four Am

stars searched have such strong elds and there is evidence that the

eld is fairly disordered The imp ortant p oint is that such stars have

an outer radiative zone and a core convection zone just like the more

massive stars Ultimately we may nd a connection in the buildup

of core magnetic elds in all upp er main sequence stars with an inner

convection zone

It is not clear at present whether the magnetic elds on the surface

of OB stars are fossil remnants from a premain sequence phase or are

elds that have b een carried through the radiative zone by circulations

Calculations for this pro cess similar to those of Charb onneau Mac

Gregor remain to b e done and are required to demonstrate that

this latter pro cess is really p ossible Such a transp ort mechanism has

to remain an assumption at this stage

B Winddriving

In the standard version of the fast magnetic rotator theory Hart

mann MacGregor Lamers Cassinelli chapter the

magnetic eld is calculated selfconsistently and turns out to b e radial

close to the stellar surface and then starts b ending at the critical p oint

where the radial Alfvenvelocity the lo cal corotation velocity and the

wind velocity all coincide This pro duces a long lever arm for the loss

of angular momentum and is the essence of the criticism of Nerney

Suess against the mo del The spindown of fast magnetic rota

tor WolfRayet stars is a topic addressed by Poe Friend Cassinelli

where it is assumed that the eld is radial close to the stellar

surface

However the magnetically driven winds do not require the mag

netic eld to b e radial near the surface In a convection zone near the

surface it may b e plausible to assume that the magnetic eld is nearly

isotropic in its turbulent character b elow the region where the wind

gets started and radial in the wind zone near to the star leaving the

radial magnetic eld lines dominant for as long as the ow velocity is

b elow the radial Alfvensp eed We note that the surface magnetic eld

structure on the Sun is mostly radial at the level where we can observe

it and we can only infer its structure b elow the surface from detailed

mo delling

However in a star where the outer layers are radiative it is not

clear at all that the magnetic eld is initially radial Even the slightest

dierential rotation will tend to make a magnetic eld which originates in the central convective region to b e tangential Even the acceleration

PL BIERMANN

into a wind do es not obviously overturn this tendency completely we

just do not know In fact the recent mo del calculations of Wolf Rayet

star winds by Kato Ib en suggest that the critical p oint of the

wind where the wind b ecomes sup ersonic may b e already inside the

photosphere and so it is reasonable to supp ose that the other critical

p oint where the wind sp eed exceeds the lo cal radial Alfven velocity

may also b e inside the star if there is such a p oint at all the radial

ow velocity may b e faster than the radial Alfvenvelocity throughout

Similarly the arguments by Lucy Abb ott which discuss the

eect of multiple scattering in radiation driving also lead to an eective

initial acceleration of the wind which may lead to a fast wind already

near the photosphere

In the following we prop ose to discuss such a wind generalizing

from Parker and Weber Davis As in Weber Davis

we limit ourselves here to the equatorial region We use magneto

hydrodynamics and Maxwells equations and thus have for the angular

momentum transp ort L p er unit mass

J

B

r

r B const L r v

J

v

r

Here the comp onents of the magnetic eld are B and B and the

r

comp onents of the wind velocity are v and v while is the density

r

and the index a refers to a reference radius which may or may not

corresp ond to the stellar surface

Introducing for the magnetic ux

F r B r B const

B r r a

a

with ux freezing

r v B v B F const

r r B

and mass ux

M r v r v const

r a r a

a

the tangential velocity can b e written as

F F L

J

B B

v

r L v r v

M M

J r r

The angular momentum loss can b e written as

L r

J

a

where denes the lo cation from where the angular momentum loss is

actually o ccurring If there is no Alfvencritical p oint outside the star then obviously

STELLAR WINDS SNe and CRs

We note that the term F M v r app earing in the expression for

r

B

the tangential velocity can b e rewritten with the surface radial Alfven

Mach number

M v v

Ar a r a Ar a

where

v B

Ar r

as

F r v

r a

a

B

v r M v r M

M

r r

Ar a Ar

Similarly we have the relationship

r v F

r a

B

L v M r M v

M

J r r

a

Ar Ar a

Thus the tangential velocity can b e written as

L v

J r a

v

v r M M

r

Ar a Ar

We can reasonably assume that the radial velocity is steadily in

creasing with radius The tangential velocity should neither b e negative

nor exceed the rotational velocity of the star itself and so we derive

the conditions

M

Ar a

and

M

Ar a

for the conditions envisaged here that there is no Alfvencritical p oint

outside the star These conditions translate into

v

r a

M v

r

Ar a

and

M

Ar

It follows eg that M r asymptotically Ar

PL BIERMANN

We dene U and U and obtain

M

r

a

U

r

and

U

M

M

Ar

The tangential magnetic eld can then b e written as

r F

B

a

B

r v r M

r

Ar

and is thus also without change of sign outside the star It is easy to

verify that no magnetic ux is transp orted to innity see Parker

The radial momentum equation is

d c

s

v v

r r

dr v

r

GM F N c

r ad T

s

r r m c

p

v

d

r B

r r dr

where indicates the nonadiabacy of the ow

dP dP

along r adiabatic

d d

We will adopt for simplicity in the following

In this equation the radiation force ux F on b oth lines and

r ad

continuum is given with the correction factor N over the Thompson

crosssection also including the eect due to the chemical element com

p osition b eing dierent from pure hydrogen this factor N dep ends on

the physical state of the gas The adiabatic sp eed of sound is c We

s

here pro ceed to evaluate the last term with the expressions already

derived and discuss it in the context of the momentum equation

The gradient term of r B in the momentum equation can then

b e rewritten as on the right hand side of the momentum equation

d

r B

r dr

d r v U

a r a

v v

r r

dr M v v U

r a r

Ar a M

v r v U

r a a r a

r

a

M r v U M v

r r

Ar a M Ar a

STELLAR WINDS SNe and CRs

A second term on the right hand side of the momentum equation

is the centrifugal force which can b e written as

v

r v

a r a

r

a

r r U M v

r

M Ar a

The last term in brackets never go es through zero outside the star

b ecause of the conditions we have set ab ove for and M Comparing

Ar a

now the centrifugal term with the second term from the gradient of the

tangential magnetic eld we note that for dv dr approaching these

r

two terms dier by the factor

v U

r M

M

Ar a

U v

r a

where M If v v exceeds a value of at large radius r

r r a

Ar a

then the centrifugal term which provides some acceleration dominates

over the other term at large r Here we have ignored all the terms of

order unity that approach unity with b oth the radius r and the radial

velocity v b ecoming large Even close to the star the centrifugal force

r

may dominate

The rst term in the gradient of the tangential magnetic eld is

more interesting however This term b ecomes a summand to the var

d

v and has there on the left hand side a ious terms multiplying v

r r

dr

minus sign and is to b e compared with unity in the case that we are

already at sup ersonic sp eed This may b e attained through radiation

driving with multiple scattering

We dene an AlfvenMachnumber with resp ect to the total mag

netic eld with

v

r

M

A

B B

r

Generally we have obviously

M M

A Ar

With this generalized AlfvenMachnumber we can rewrite the entire

d

v in a simple form factor to v

r r

dr

M

Ar

M

Ar

M M

s

A

where M is the sonic Mach number for the radial ow velocity Critical

s

p oints app ear whenever this expression go es through zero or a singu

larity This expression is easily seen to b e equivalent to eq of Hart

mann MacGregor they however nd a magnetic eld which

is initially radial and neglected radiative forces This also shows that

PL BIERMANN

we have the critical p oint of Weber Davis again for

2

M

s

when and if M but we have in our case another critical p oint at

Ar

M which is the fast magnetosonic p oint see Hartmann Mac

A

Gregor In order to make a realistic judgement on this question

we would have to combine the mo del of Kato Ib en or a de

tailed calculation of multiple scattering Lucy Abb ott to drive

winds initially with a realistic treatment of the magnetic eld inside

the star so that we can derive the range of p ossible prop erties of the

magnetic eld near the surface of the star

However there are a few general conclusions one can draw already

If the magnetic eld inside the star b egins as a dominantly tangential

eld then it is by no means clear whether there is any p oint at which

we have M either inside or outside the star going outwards the

Ar

unwinding of the magnetic eld is intimately coupled to the initially

weak radial ow and so M may hold throughout In that case

Ar

the term derived from the radial gradient of the tangential magnetic

eld go es through unity only when the generalized AlfvenMach num

b er go es through unity If as we argued M p ossible throughout

Ar

then again going outwards with radius we start with a low generalized

AlfvenMach numbers M either sup ersonic or subsonic ow the en

A

tire expression is negative matching the negative gravitational force on

the right hand side Far out b oth sonic and generalized AlfvenMach

numbers are large and so the expression is p ositive It follows that the

expression has to go through zero The condition M here is the

A

fast magnetosonic p oint and the radial velocity there corresp onds to

the Michelvelocity Or in other words the radial velocity is equal to

the total Alfvenvelocity

We know from observations that the winds in OB and Wolf Rayet

stars are very strongly sup ersonic using our mo del we deduce that the

radial velocity is weakly sup erAlfvenicwith resp ect to the tangential

dominant magnetic eld comp onent and so we exp ect far outside the

star a conguration where M M and M but this

s Ar A



latter condition may not necessarily b e satised by much This means

that the expression is near to unity and somewhat smaller than unity

This entails the condition that the right hand side has to b e p ositive

which is readily interpreted as p ossibly arising from line radiation b e

cause that term is the only one which has the same radial dep endence

as the gravitational force Comparing then the eect of line driving

Lucy Solomon Castor et al with and without such a

magnetic eld the net eect is an amplication in the sense that for

M the velocity gradient is asymptotically increased by

Ar

M A

STELLAR WINDS SNe and CRs

For M close to unity this factor can b e arbitrarily large and

A

thus illustrates the amplication p ossible from the pressure gradient

of a tangential magnetic eld Obviously such a large amplication is

present only for a small radial region and so the nal wind velocity

can still likely b e not much more than the tangential Alfvenvelocity

see b elow We thus argue that there is a magnetic eld conguration

where the eld is mostly tangential already inside the star and remains

mostly tangential outside the star where the initial acceleration of the

wind is done by the opacity mechanism discussed by Kato Ib en

or by multiple scattering Lucy Abb ott in a linedriving mo dus

outside the star we have a linedriven wind but the amplication by

the eect of the pressure gradient of the tangential magnetic eld really

pro duces the large momentum in the wind

To see the prop erties of the equations b etter we simplify by drop

ping all right hand terms in the dierential equation except for the grav

itational and the radiative force and introduce characteristic length

and velocity scales

v v GM

r a r a

r

v v r

Ar a a

r a

and

r v

a Ar a

v v

r a

v v

r a r a

which is easily recognized as the Michel velocity Hartmann Mac

Gregor This equation corrects a misprint in the corresp onding

expression in pap er CR I I in that the second exp onent is instead

of These normalizations lead to the same overall dimensionless

form of the momentum equation as in Owocki since we have to

multiply the entire eq with

r r

r v GM

The Eddington luminosity is given by

GM m c

p

L

edd

T

where is the Thompson cross section In the approximation that

T

U U and r v the Michel velocity corresp onds to the

M r

overall Alfvenvelocity

Next we write down the whole dierential wind equation in dimen sionless form

PL BIERMANN

dy M

Ar

M

Ar

d x M M

s

A

v r

r a a

c r

a

sa

r v GM

r

LN

L

edd

v r

r a a

r

a

r U M v

r

M Ar a

U v v

r a r a

U v M M v GM

M r r

Ar a Ar a

Here we have to note that in a full treatment of this equation the

temp erature enters in the sp eed of sound here approximated by an

adiabatic law with gas constant the state of ionization enters into

the factor N which can b e highly space dep endent see eg Lucy

Abb ott and that hence we can discuss here only a very simplied

form of this equation We approximate this expression in several steps

in order to nally write down only two limiting forms

In the approximation that the sonic and the radial AlfvenicMach

number are b oth large already on the surface the left hand side sim

plies to

dy

d x M

A

showing only the fast magnetosonic p oint as a remaining singularity

At that p oint the right hand side has to go from negative to p ositive

which could happ en in several ways First of all the pressure term is

clearly unimp ortant since we already neglected the subsonic regime

The last term which derives from the centrifugal force and the second

part of the magnetic eld pressure gradient is asymptotically p ositive

on the surface this term may b e negative Writing

E

M

Ar a

with the condition

E M

Ar a

we nd the condition on the surface

M

Ar a

E

M

Ar a

for the last term in eq to b e negative on the surface It thus

dep ends on a detailed consideration of the inner structure of the star

STELLAR WINDS SNe and CRs

and its rotation law to determine whether this term can b e suciently

large and negative on the surface to comp ensate the radiative driving

term and so to lo cate the fast magnetosonic p oint

Another p ossibility is that the radial dep endence of the radiative

driving term ie the radial variation of the factor N determines the

lo cation of the fast magnetosonic p oint In that case we may b e allowed

to neglect the centrifugal and the remaining term of the magnetic pres

sure gradient

The dierential wind equation then reads nally in dimensionless

length x and velocity y

dy

f

r ad

y d x

with

NL

f

r ad

L

edd

We note that the radial dep endence of the magnetic eld strength

leads asymptotically to

U M

M Ar

M y y

Ar

M U

A

U in the limit that U

M

Clearly we require then that the radiative force dominates over

gravity by however little to obtain the correct sign of the right hand

side thus f We assume here for illustration that N is constant

r ad

with r

The analytic solution is

f y y

r ad

i

y y x x

i i

where the index i refers to an initial state For y y and x x

i i

the limiting solution is

r

f y

r ad

y x

i i

We note right away that b oth eects contribute to a faster wind

Thus the analytic solution yields then two extreme p ossibilities either

NL

v v

r 1 e

L

edd or

PL BIERMANN

p

v r

Ar a a

v v

r 1

v v

r a r a

where we have used the denition

GM

v

e

r

a

for the surface escap e velocity from the star Obviously we require

that

v r

Ar a a

v v

r a r a

in order to have a nal velocity ab ove the Michel velocity We note

that

r B U

M

v B U

r r

for any r and also on the surface and thus the condition is likely to b e

fullled for a magnetic eld conguration which is highly tangential at

the stellar surface By the same condition the initial surface velocity

is b elow the Michel velocity except for the factor which is of order

unity This is necessary for the ow to have a transition through the

fast magnetosonic p oint This sharp ens the approximations introduced

ab ove We note that in the magnetic driving case the nal velocity far

outside can also b e written as

p

v r

Ar a a

v v

r 1 e

v v

r a e

p

where b oth factors to v are less than unity in our approximation

e

and so the case of pure magnetic driving in this approximation may

yield lower velocities than radiation driving at large distances from the

star in this very simple rst approximation The transition b etween

the two extreme cases is given by

NL r v

a Ar a

v v L

r a e edd

We have thus in this approximation two p ossible extreme solutions

First for small magnetic eld the Alfvenvelocity drops out and the

line driving is the regulating agency second for large magnetic eld

the terminal wind velocity is not far from the tangential Alfvensp eed

and the line driving eect is imp ortant but not dominant in that it

provides the source of momentum to b e amplied This latter picture

is the concept we prop ose to explain the momentum in the winds of

Wolf Rayet stars This is dierent from the solutions of Hartmann

STELLAR WINDS SNe and CRs

MacGregor in the sense that in their wind solutions the main

acceleration of the wind is b etween the slow magnetosonic p oint and the

Alfvenp oint b oth of which are not outside the star in the case which

we discuss our solutions are similar however in that b oth in their

case as here the terminal velocity of the wind in the case of interest is

not far from the the total Alfvenvelocity In this picture the angular

momentum loss of the star refers to a characteristic level inside the star

and so the criticism of Nerney Suess is countered the angular

momentum loss of the star through mass loss is reduced

Introducing then the radiative driving in the form of Castor et al

and using the nomenclature of Owocki we have for the

radiation driving the following form

dy NL

C y

C AK

L dx

edd

where C is related to the mass loss rate Owocki there eq

C AK

and LL describ es the eect of the continuum radiation

edd

and thus in a second approximation

dy dy

C y

C AK

y d x dx

dy

This replaces then eq of Owocki Here C y

C AK

dx

describ es the driving by radiative forces in the approximation by Cas

tor et al and describ es the spatial dep endence of N ab ove

This shows that we have a critical p oint where y and where

dy

We have to ask here whether we have a C y

C AK

dx

critical p oint at all where M Consider the case of low magnetic

A

eld Then the initial surface ow velocity may well b e sup erAlfvenic

and so y and if in fact y then the magnetic eld can b e

a a

neglected However if the initial velocity is subAlfvenicand the nal

radiationdriven wind is sup erAlfvenicthen magnetic driving certainly

cannot b e neglected These conditions can b e written as

v v v

r a r a r a

v v r

Ar a a

and

r

v v v v

e r a r a C AK

v v v r

r a Ar a a

If these two conditions are fullled then we certainly have a critical

p oint where y Since r v prop ortional to the ratio of the

a r a tangential over radial comp onent of the magnetic eld on the reference

PL BIERMANN

level of the star see eq a suciently dominant tangential eld

will always ensure that the rst condition is fullled Strong wind

driving corresp onds to close to unity and so in many cases the second

condition will also b e fullled In the case that the rst condition

is fullled but not the second the eects of the magnetic eld also

cannot b e neglected but a full solution is required to discuss this at

the required depth A full solution of this equation remains to

b e done

Using the observed wind velocities then gives an estimate for the

Alfvenvelocity and thus implies assuming our wind driving theory

to b e correct that the magnetic eld is quite high and of the order

of Gauss at the ducial radius of cm Or to turn the argument

around the magnetic eld strengths implied by our cosmic ray argu

ments provide a p ossible explanation for the origin of the momentum

of Wolf Rayet star winds On the surface of the star the magnetic eld

strength implied is of order a few thousand Gauss quite easily within

the limits implied by the surface virial theorem Also the surface of

the star may not rotate as fast but the inside could still rotate at an

angular velocity corresp onding to near critical at the surface We em

phasize that here our prop osed wind driving theory if true provides

an indep endent argument on the magnetic eld strength on the surface

of Wolf Rayet stars If it can b e shown to b e the correct physical in

terpretation then these magnetic eld strengths follow On the other

hand if observations show such magnetic eld strengths to b e to o high

then b oth the wind theory prop osed here as well some of the cosmic

ray arguments may fail

C Radio emission from stars

In this section we derive the basic expressions for the luminosity

sp ectrum and time dep endence of the nonthermal radio emission from

single spherical sho cks in the winds of single massive stars In the

subsequent sections we then use these expressions to discuss radiosu

p ernovae young radio sup ernova remnants in starburst galaxies Wolf

Rayet stars and OB stars The radio emission of the nova GK Per

Seaquist et al remains to b e discussed in the detail required to

match it with what we know ab out cataclysmic variables close binary

star systems with an accretion disk around a white dwarf Biermann

Strom Falcke

Electrons suer massive losses due to Synchrotron radiation and

so they can achieve only energies up to that p oint where the accelera

tion and loss times b ecome equal this is a strong function of latitude

since the magnetic eld varies rather strongly with latitude The sho ck

transfers a fraction of the bulk ow energy into relativistic electrons The emissivity of an electron p opulation with the sp ectrum

STELLAR WINDS SNe and CRs

p

N d C d

is given by Rybicki Lightman in cgs units for p by

CB erg sec cm Hz

The case of a wind sp eed equal to the sho ck velocity and the limit

of a strong sho ck in a gas of adiabatic index leads to an optically

thin synchrotron sp ectrum of and thus p The corre

sp onding expressions for p are given in pap er CR I I Similarly

the absorption co ecient for synchrotron self absorption is for p

CB cm

sy n

In all these expressions we have averaged the asp ect angle We

will use here for illustration the limit of large sho ck sp eeds for the

discussion of the radiosup ernovae where the radio sp ectral index is

The freefree opacity is given by

T n cm

f f

e e

in an approximation originally derived by Altenho et al and

widely disseminated by Mezger Henderson Here T and n are

e e

the electron temp erature and density and the approximation has b een

used that the ion and electron density are equal and that the eective

charge of the ions is Z We note that for cosmic abundances

we have the following approximations Schmutzler assuming full

ionization m n n n and n n

H e i

where n is the total Hydrogen density particles of mass m p er cc

H

and n the ion density With these approximations for full ionization the

i

freefree opacity would increase a factor of two over the approximation

by Altenho et al which however is comp ensated by the

incomplete ionization see again the calculations by Schmutzler

in the photon ionized regions here winds near massive stars In the

following we will use the approximation by Altenho et al replacing

n n

e

The synchrotron luminosity in the optically thin limit is then given

by an integration over the emitting volume which we take in the con

text of our simplied picture to b e a spherical shell of thickness r

The radio emission only arises from that part of the shell where the

sho ck velocity is larger than the lo cal Alfven velocity and where the

synchrotron loss time is longer than the lo cal acceleration time These

conditions can lead to a restricted range in latitude for the emission

see eg Nath Biermann b Biermann Strom Falcke

PL BIERMANN

The luminosity is then given by the integral over the latitude dep endent

emissivity Here we have to normalize the cosmic ray electron density

to the sho ck energy density using our induced latitude dep endence

We will concentrate on the case when the energetic particle density is

stronger near the equator and discuss the opp osite case briey at the

end Using a lower b ound for the electron sp ectrum much b elow the

rest mass energy and an upp er b ound much ab ove that value we have

for the constant C then the expression

C C

o

with where refers to that latitude where the latitude

dep endent acceleration breaks down

U

c

from our argument ab out the knee energy see section of pap er CR

I for protons and other nuclei At the equator the energy density of

the electrons can b e written as

C p U m c

o e

where we use just the relativistic part of the distribution function and

incorp orate the uncertainty on the existence and strength of any sub

relativistic part of the electron distribution function in the factor

The integration over latitudes leads to an integral correction factor of

p

B where B z z is the Betafunction and p the p owerlaw

index of the electron distribution function For the p owerlaw index

used here p this integral is very close to in value We also

have

M

r V

W

Assuming that all latitudes contribute up to the nal expres



sion for the luminosity is then

L nth erg secHz

M

B r U

V

W

Here the mass loss is in units of M yr the unp erturb ed

magnetic eld strength at the reference radius of cm in units of

of Gauss the wind and the sho ck velocity in units of c and as

reference frequency we use GHz

Here we note that this emission might b ecome optically thick b oth

to freefree absorption in the lower temp erature region outside the

STELLAR WINDS SNe and CRs

sho ck or to to Synchrotron self absorption inside the sho cked region In

the approximation that we use the equatorial region in the slab mo del

ie direct central axis radial integration we obtain for the critical

radius r where the freefree absorption has optical thickness unity

f f

M

r cm

f f

V

W

We have used here the analytical approximation for the freefree

opacity introduced ab ove and assumed a temp erature of T K

e

Similarly for synchrotron self absorption the critical radius r

sy n

is given for p

r cm

sy n

M

U B

V

W

The maximum synchrotron luminosity is given by the dominant

absorption pro cess which one is stronger is given by comparing the

relevant radii in the numerical example synchrotron self absorption

happ ens to b e stronger Freefree absorption is dominant for p

if

M

U B

V

W

We note that the parameter might well b e quite low

In the following we give then the maximum luminosities calculated

by using the prop er optical depth for a central axis approximation in

a slab geometry for the radiative transfer at the equator this gives an

additional factor of ab out see b elow However for the total emis

sion we do integrate prop erly over all latitudes while for the absorption

we approximate by using the equatorial values This is a fair approxi

mation since b oth emission and absorption decrease towards the p oles

but the absorption decreases even faster The exact numerical value in

our approximation is used here

In the case that freefree absorption dominates the maximum lu

minosity is then given for p by

L nth erg sec Hz

M

U B

V

W

In the case that Synchrotron self absorption dominates these max imum luminosities are given by

PL BIERMANN

L nth erg sec Hz

M

U B

V

W

Here we do not consider mixed cases

Obviously the ratio of mass loss rate and wind velocity can b e

dierent by many orders of magnitude among predecessor sup ernova

stars and their numerical values will have to b e argued on the basis of

observations

It is useful to also calculate the thermal radio emission using the

same standard parameters then adjust our parameters to a realistic

range and then compare the nonthermal luminosities The thermal

emission can b e prop erly integrated allowing for the sphericity of the

wind structure Biermann et al to give

L th B T r

f f

With our standard parameters this luminosity in a steady wind

M

L th erg secHz

V

W

is weakly dep endent on electron temp erature

In the following we make a number of consistency checks on the

basic notions used ab ove

First we note that sho cks can only exist when the sho ck velocity

is larger than the Alfven velocity Since in our wind driving theory

developed ab ove the wind velocity is just a little larger than the Alfven

velocity itself this implies that the sho ck velocity in the frame of the

ow has to b e at least the same velocity as the wind and so within

our analytical approximations we have the condition that always

U V

W

where the limit of the equality corresp onds to a sp ectral index for the

Synchrotron emission of and in the limit of large sho ck velocity

to Here we have to note that the Alfvenvelocity is colatitude

dep endent and is prop ortional to sin Thus even for lower sho ck

sp eeds there is a latitude range where a sho ck can b e formed but then

the luminosity is very much reduced

Second we have to check that the Synchrotron loss time is larger

than the acceleration time b ecause otherwise we would not have any

electrons at the appropiate relativistic energies This condition can b e rewritten as

STELLAR WINDS SNe and CRs

U r B



This is the condition at the equator and the condition gets weaker

at higher latitudes This makes it obvious that for reasonable ranges

of the parameters acceleration can succeed to the required electron

energies It also shows that at higher frequencies or smaller radii the

condition would fail At smaller radii the emission is usually optically

thick see ab ove and higher frequencies are as yet dicult to observe

at the required sensitivity The implied high frequency cuto in the

observed synchrotron sp ectra would however b e an imp ortant clue

Third we have to check whether the implication that the sho ck

sp eeds are typically similar to the wind sp eeds is supp orted by data

on Wolf Rayet stars and their theoretical understanding The wind

calculations with sho cks suggest that the typical sho ck velocities are

indeed of order the wind sp eed itself or somewhat higher Owocki et

al and so we exp ect a range in radio sp ectral indices of

to or steep er if the sho cks are not strong ie if U U

We note that the actual value of the nonthermal luminosity is changed

only mo derately in this range of sp ectral indices

Fourth we have to discuss optical thickness eects in more detail

A sho ck travels from the region inside of where freefree absorption

dominates through this region to the outside The emission then is rst

weak and has a steep sp ectrum due to the strong frequency dep endence

of the freefree absorption and the exp onential cuto induced then

approaches a p eak in emission with a sp ectral index approaching

to near ab out or steep er as discussed ab ove and then b ecomes

weaker with the optically thin sp ectrum At the lo cation where our ray

encounters the sho ck the temp erature of the gas increases drastically

and so the dierential optical depth for freefree absorption go es to zero

Hence we have the simple case that we have emission inside the sho ck

and absorption outside We limit ourselves to the central axis as a rst

approximation and also neglect the spatial variation of the emission

itself see ab ove This is equivalent to pure screenlike absorption

however with a screen which extends from the sho ck to the outside

and so is variable The radio luminosity is given by using freefree

absorption

L nth no abs L nth e

We have the sp ectral index in the optically thin regime

thin

and the time dep endence of the sp ectral index given by

t t

thin

The sp ectral index is p ositive and very steep at rst and then go es through zero to b ecome slowly negative approaching asymptotically the

PL BIERMANN

optically thin sp ectral index The nonthermal luminosity considered as

a function of time sharply rises at rst then p eaks at optical depth

obviously strongly dep endent on frequency and nally drops

thin

thin

o as t

d l n L nth

t t

thin

d l n t

Here we see that the time dep endence of the ux density at a given

frequency and the time dep endence of the sp ectral index are closely

related The time of luminosity maximum dep ends on frequency as

t

max L

from the frequency dep endence of the optical depth

t

These relationships can b e used to check on the imp ortance of free

free absorption in our approximations

Synchrotron self absorption is due to internal absorption inside the

shell and so again in the central axis approximation we have

e

L nth L nth no abs

This then results in the time dep endence for the sp ectral index of

thin thin

e

with

t

The time dep endence of the double logarithmic derivative of lumi

nosity on time is exactly the same

d l n L nth

d l n t

It follows that the time of maximum dep ends on frequency as

t

max L

This is a characteristic feature for synchrotron self absorption in

the approximation used and well known from radio quasars and diers from the case considered ab ove for freefree absorption

STELLAR WINDS SNe and CRs

Thus the true maximum of the luminosity is given by an optical

depth less than unity which gives a correction factor to the luminosities

introduced ab ove eqs and of ab out we have corrected

the luminosities introduced there for this factor

Finally we have to comment on the sign of the magnetic eld We

have used here throughout the assumption that the magnetic eld is

oriented such that the drifts are towards the equator for the particles

considered Clearly since we consider stars with a magnetic eld driven

by turbulent convection in a rotating system we can exp ect that there

are sign reversals of the magnetic eld just as on the Sun For the

other sign of the magnetic eld and the other drift direction there is by

many p owers of ten less nonthermal emission and so it is to b e exp ected

that at any given time we should detect at most half of the stars in

nonthermal radio emission o ccasionally we might even catch a sho ck

travelling through the region in the wind where the sign is reversing

itself b ecause the wind mirrors the time history of the star in terms

of magnetic eld In the case that the sho ck travels through a layer

b ounded by magnetic eld reversals b oth inside and outside then it

b ecomes imp ortant to ask what the time scale of particle acceleration

is relative to the time scale of traversing this layer particle acceleration

at high energies may b e severely limited if such layers are thin

In the following we will rst discuss the radio observations of Wolf

Rayet stars then OB stars and nally sup ernovae

The theoretical luminosities derived can b e compared with the ob

servations of Wolf Rayet stars which yield Abb ott et al non

thermal luminosities up to ab out erg secHz at GHz and ther

mal luminosities up to erg secHz Since the most imp ortant

parameter that enters here is the density of the wind or in terms of

wind parameters the ratio of the mass loss to the wind sp eed M V

W

we induce that the most extreme stars have a higher value for M V

W

by This translates into a higher nonthermal emission as well by

a factor of to give erg secHz U B using the

case when freefree absorption dominates for synchrotron self absorp

tion the corresp onding luminosity is very nearly the same This is

very much more than the observed nonthermal luminosities and sug

gests that there is a limiting factor We implicitly assume here that

some of the massive stars that explo ded as the observed sup ernovae

either were Wolf Rayet stars b efore their explosion or that their prop

erties were not signicantly dierent since we derived the wind density

from the timing of the lightcurve ab ove and the sho ck velocity b oth

from the mo dels of Owocki et al and the argument that the

sho ck velocity b e larger than the Alfven velocity which in turn we

argued is not very much lower than the wind velocity the only other

imp ortant parameter is the magnetic eld strength and that we as

sume then to b e of similar magnitude Since Wolf Rayet stars do not

PL BIERMANN

distinguish themselves from stars that somewhat later in life explo de as

sup ernovae we can use all the same parameters except for the sho ck

sp eed Then the only parameter left is the highly uncertain eciency

of electron injection

We can thus ask whether the eciency might dep end on sho ck

sp eed The sho ck sp eeds in sup ernovae are of order c or larger

while those in Wolf Rayet star winds are of order c or less Some

where b etween these sp eeds there app ears to b e a critical sho ck sp eed

at which the character of electron injection changes Here we do not

wish to discuss injection but merely note that the data suggest ecien



cies of order for the sho ck sp eeds thought to b e normal in

Wolf Rayet star winds and of order for sup ernova explosions

The data thus suggest that the injection of electrons into the diusive

sho ck acceleration app ears to exhibit a step function prop erty at a crit

ical sho ck velocity Comparing normal sup ernova remnants the nova

GK Per and the sources discussed ab ove a natural choice is a critical

AlfvenicMachnumber which we can only estimate to b e in the range

of to If true this might b e imp ortant also for electron injection in

quasars which also exhibit a dichotomy into radioloud and radioweak

ob jects In pap er CR I I I and b elow we suggest that this concept leads

to an estimate for the electronproton ratio of the lower energy cosmic

rays consistent with observations

Now we can go back and ask again whether freefree absorption or

synchrotron self absorption dominates in the various cases considered

Clearly when the eciency for electron injection is very small then

freefree absorption always dominates and so the maximum luminosi

ties during a radio variability episo de of a Wolf Rayet or OB star

should b e frequency dep endent Also for slow winds in sup ernova pre

decessor stars again freefree absorption will dominate normally eg

in the two examples used ab ove On the other hand for fast winds

of the sup ernova predecessor stars as in the case of a Wolf Rayet star

explo ding synchrotron self absorption is likely to b e stronger than

freefree absorption and then the maximum luminosities at dierent

radio frequencies should b e indep endent of frequency This is a testable

prediction since it relates radio and optical prop erties of a young su

p ernova

D OB stars

The theoretical luminosities derived can b e compared with the ob

servations of OB stars which yield Bieging et al nonthermal

luminosities up to ab out erg secHz at GHz and thermal

luminosities up to erg secHz Since the most imp ortant pa

rameter that enters here is the density of the wind or in terms of wind

parameters the ratio of the mass loss to the wind sp eed M V we

W

induce that the most extreme stars have a higher value for M V by W

STELLAR WINDS SNe and CRs

This translates into a higher nonthermal emission as well by a

factor of to give erg secHz U B This is very

much more than the observed nonthermal luminosities Since the sho ck

prop erties are likely to b e similar to Wolf Rayet stars this is again con

sistent with the idea that the injection eciency of electrons might b e

quite low in conjunction with magnetic eld values not to o far from

what we argued to b e valid for Wolf Rayet stars

We can also compare the detection statistics and observed sp ectral

indices Since in a variability episo de a sho ck comes from b elow through

the region of optical thickness near unity the nonthermal emission

increases rapidly to its maximum when its sp ectral index is

max

thin

which is approximately Thereafter as the luminosity more slowly

decreases the sp ectral index gradually approaches the optically thin in

dex of or slightly steep er Therefore we exp ect the sp ectral index

distribution of detected sources to b e a broad distribution from near

to near this is what has b een found Bieging et al

Because of the two p ossible signs of the magnetic eld orientation and

the asso ciated drift energy gains for particles we also exp ect that at

any given time at most half of all sources are detectable with nonther

mal emission even at extreme sensitivity This is consistent with the

observations We predict a similar b ehaviour for Wolf Rayet stars

Here we have to ask how the magnetic elds can p enetrate the ra

diative region from b elow This question cannot presently b e tackled

by simulations on a very large computer yet but it is likely that the cir

culations induced by rotation transp ort magnetic elds to the surface

where even a rather slight dierential rotation draws out the magnetic

eld into a mostly tangential conguration In this case clearly the

origin of the momentum of the wind can b e readily accounted for from

line driving Lucy Solomon Castor et al and so we

susp ect that the magnetic driving adds only little which is the other

case discussed ab ove near the end of the wind section

We can also compare with the only existing theory to explain the

nonthermal radio emission of single massive stars with winds by White

Whites theory is based on a concept involving a large num

b er of sho cks and do es not explain the data as already demonstrated

by Bieging et al in terms of i radio sp ectral index ii time

variability nor iii of the statistics of detection As a consequence any

estimate of the strengths of magnetic elds based on Whites theory

is in doubt see Bieging et al The theory of diusive particle

acceleration by an ensemble of sho ck waves has b een prop erly derived

by Schneider and remains to b e compared with the data of

stars Our theory is based on using single sho cks and readily provides

PL BIERMANN

sp ectral indices in the entire range that is observed it explains the time

variability and easily accounts for a fair fraction of undetected sources

Sho cks in stellar winds as a consequence of sup ernova explosions

give rise to ray emission from hadronic interactions Berezinsky

Ptuskin this latter conclusion was reached also for normal sho cks

in stellar winds by Chen White b and White Chen

In the case of sup ernova explosions it is imp ortant as a check that

the resulting predicted ray pro duction scales with the square of the

winddensity and thus with M V Berezinskii et al eq

W

in VI Ix

The binary system WR has b een detected by GRO Hermsen

et al seminar at the GRO symp osium at Maryland OSSE

and BATSE instruments conrming a detailed prediction by Eichler

Usov which is based on a mo del of colliding wind sho cks

This latter agreement of observation and prediction is a consistency

check on the concept that freshly accelerated protons pro duce high

energy photons from hadronic interaction as opp osed to purely leptonic

pro cesses

E Radio emission from radiosup ernovae

The adopted value for the magnetic eld however was derived

from the notion that the subsequent sup ernova sho cks accelerate parti

cles to extremely high energies This argument can b e checked with the

observations of those sup ernovae of which the radio emission in the wind

was really observed ve sources discussed by Weiler and colleagues in

a number of pap ers Weiler et al Panagia et

al and the sup ernova A Turtle et al Jauncey et al

StaveleySmith et al and related radio sources in starburst

galaxies However we restrict ourselves to those sup ernovae for which

we can reasonably assume that the predecessor star was indeed a star

with a strong wind and this we will do using statistical arguments in

the subsequent section

In fact our mo del can b e paraphrased as a numerical version of

Chevaliers mo del with the parameters xed In the screen ap

proximation valid for freefree absorption see ab ove for details the

time evolution of the nonthermal emission in Chevaliers mo del can b e

written as

K t

2

L nth K t e

with and to b e tted to the data For fast strong sho cks our

mo del predicts that and and for slower sho cks

that still but larger in number for U V

W

for instance This is consistent with the detailed ts for three

sup ernovae Weiler et al which app ear to arise from stars with

STELLAR WINDS SNe and CRs

stellar winds Using the numbers from the t using Chevaliers mo del

which requires the averages are hi

h i h i h i and

h i from these somewhat sparse data

For SN J the data clearly cover a suciently large time to

test this numerical mo del in more detail Weiler et al nd that

this simple mo del in its screen approximation do es not provide a go o d

t to the early ep o chs We susp ect similar to Weiler et al that this

lack of a go o d t can b e traced to the simplication that we consider

the external absorption as a simple screen and disregard the lateral

structure Further p ossible reasons for a failure to strictly adhere to

the simplied mo del are the following a The presho ck wind may

not b e smo oth in its radial b ehaviour there might have b een a weaker

sho ck running through earlier which nevertheless can disturb the radial

density prole see the calculations by MacFarlane Cassinelli

b Mixing b etween freefree absorption outside the sho ck region and

synchrotron selfabsorption inside the sho ck region c The structure

of acceleration in its latitude dep endence is considered here only for

the acceleration of particles see pap er CR I but not for absorption

and radiative transfer Given a very detailed multifrequency data set

it would b e interesting to mo del the data fully

In fact we can use the numerical values for the radii for maximum

luminosity derived ab ove to obtain the presho ck wind density which

is prop ortional to M V With the data given by Weiler et al

W

this yields for the sup ernova C

M

C U

V

W

and for the sup ernova K

M

K U

V

W

We note here that the nomenclature for these sup ernovae has changed

b etween Weiler et al and Weiler et al we use here the

more recent version

Clearly these numbers tell us that the predecessor stars had a slow

wind and thus were probably red sup ergiants see Weiler et al

This then implies for sho ck sp eeds of c and freefree

absorption b eing dominant that the nonthermal luminosities at GHz

exp ected versus observed are

L C versus erg sec Hz

max and

PL BIERMANN

L K versus erg sec Hz

max

b oth for the assumed strength of the magnetic eld Since the lumi

nosity is prop ortional to the magnetic eld strength to the p ower

and directly prop ortional to the electron eciency parameter we de

rive thus a lower limit to the magnetic eld given an upp er limit on

Since is unlikely to b e surpassed by much using the ratio

of the luminosities exp ectedobserved of the two cases ab ove yields an

estimated lower limit to the magnetic eld strength at our reference

radius of

B U Gauss

This implies a strong lower limit to the magnetic eld from using

of Gauss The uncertainty in these estimates is clearly at

least a factor of

The argument is often made that hydrodynamic instabilities could

increase the magnetic eld strength in the p ostsho ck ow Reynolds

Chevalier in such a case the estimate given here would only

signify how much the magnetic eld has increased b ehind the sho ck

However the work by Galloway Pro ctor suggests that dy

namo time scales are not fast enough to do this As emphasized in the

summary a direct observational check on the magnetic eld strength

in stellar winds is very desirable it may b e p ossible with data from a

pulsar in a binary system with a massive early type star

For A the initial radio luminosity was indeed of order

ergsecHz and so applying our mo del with a sho ck sp eed of order

c is consistent with our exp ectation of then ergsecHz

Similarly for the new radiosup ernova J in the galaxy M the ob

served maximum radioluminosity of ergsecHz at GHz

Phillips Kulkarni is quite compatible with a reasonable sho ck

sp eed and allowing for the fact that the predecessor star was a red

giant thus V b oth optically thin time evolution and opti

W

cally thin sp ectrum are also consistent with the arguments presented

here Panagia et al

This demonstrates that strong magnetic elds also exist in the

winds of massive stars in the red part of the HertzsprungRussell di

agram where the winds are slow as argued earlier note that the

predecessor to sup ernova A was a blue sup ergiant with a fast wind

The only unknown parameter in all these predictions is the e

ciency of electron acceleration and the strength of the magnetic eld

with close to clearly close to the maximum number reasonable

leads then to the requirement that the magnetic eld strength is near

to what we assumed Gauss at cm but even higher if is very

STELLAR WINDS SNe and CRs

much less than unity This again conrms indep endently that indeed

the magnetic eld has to b e as high as argued by Cassinelli

to drive the winds and as is required to accelerate cosmic rays particles

to energies near GeV

For sup ernova predecessor stars with fast winds like Wolf Rayet

and OB stars it is of interest to ask whether the exp ected luminosity

violates the Compton limit famous from the study of radio quasars At

the Compton limit the rst order inverse Compton Xray luminosity

b ecomes equal to the synchrotron luminosity and it has b een found

from observations that compact radio quasars are close to this limit and

indeed have strong Xray emission This question can b e formulated

as a limit to the brightness temp erature of the radio source using here

synchrotron self absorption which then gives the limit

M

U B

V

W

For freefree absorption the limit is

M

U B

V

W

The rst of these conditions is almost certainly always fullled

while the second one may b e so tight as to suggest that inverse Comp

ton Xrays might b e observable This has indeed b een checked with

mo delling successfully the observed Xray sp ectra b eyond photon en

ergies of keV by Chen White a for Orion OB stars This

suggests that the inverse Compton Xray luminosity ought to b e less

usually considerably less than the Synchrotron luminosity Most of

the Xray emission from hot stars is thought to arise from those same

sho cks in freefree Xray emission which we consider for particle ac

celeration a mo delling of this was done by White Long and

MacFarlane Cassinelli

F The statistics of Wolf Rayet stars and sup ernovae

There are a variety of ways to estimate the relative frequency of

Wolf Rayet star sup ernova explosions relative to sup ernova explosions

of lower mass stars Hidayat Leitherer Massey Arman

dro Shara et al

In our Galaxy there are b etween and Wolf Rayet stars

which have an average lifetime of ab out years This gives an esti

mated o ccurrence of Wolf Rayet sup ernovae of ab out one every to

years Since the total rate of sup ernovae in our Galaxy is estimated

at ab out one every years this means that roughly one in to one in sup ernovae ought to represent the explosion of a Wolf Rayet star

PL BIERMANN

The numbers of stars on the main sequence b etween solar masses

and ab out solar masses and b etween solar masses and the upp er

end of the main sequence also ought to corresp ond to the ratio of Wolf

Rayet stars and the rest of those stars which explo de as sup ernovae

Using for the simple estimate the Salp eter mass function gives here an

estimated ratio of ab out in sup ernova events which originate from

a Wolf Rayet star in sup ernova events come from a star with a

strong wind approximately Wheeler where the change to stars

with only a weak wind is estimated to b e near a main sequence mass

of M

The mo del for the origin of the high energy p opulation of relativis

tic cosmic rays prop osed in pap er CR I and tested successfully in pap er

CR IV suggests from the energetics a ratio of ab out in assuming

the amount of energy pump ed into cosmic rays p er sup ernova to b e

the same for all kinds However here we lump all sup ernova explosions

into stellar winds together b oth with fast and slow winds

Hence in the sample of radio sup ernovae of type I I sources pre

sented by Weiler et al there ought to b e b etween none and

two events based on Wolf Rayet star explosions in the sample of radio

sources likely also to b e very young sup ernova remnants sources

with radio luminosities at GHz in the starburst galaxy M Kro

nberg et al there ought to b e ab out at least b etween and

sources which originate from a Wolf Rayet star explosion On the other

hand all of these ob jects are p ossibly explosions of stars into former

stellar winds there is evidence that the initial mass function in star

burst galaxies is biased in favor of massive stars and so the prop ortion

of the stars among them that are due to Wolf Rayet star explosions

could b e even higher than estimated here The radio luminosities are

very similar for the sources in M and the radiosup ernovae of Weiler

et al the average luminosity calculated in a variety of ways

is always in the range of erg sec Hz and erg sec Hz

tting our exp ectation see ab ove for U c rather well We

note that for the same wind density which is prop ortional to M V

W

the range of exp ected radio maximum luminosities is similar for Su

p ernova explosions into slow and fast winds on the other hand if the

mass loss rates are similar for slow and fast winds then the densities

can b e much higher in slow winds and the exp ected nonthermal radio

luminosities are much higher for slow wind stellar explosions Thus

observationally we may only detect radiosup ernovae of stars explo ding

into slow winds There is one observational signature which may b e

dicult to accurately measure For a given mass loss rate slow winds

are usually dominated by freefree absorption which leads to a fre

quency dep endent maximum luminosity see eq while fast wind

are likely to b e dominated by synchrotronselfabsorption which leads

to a frequency indep endent maximum radio luminosity In Weiler et

STELLAR WINDS SNe and CRs

al the data for the sup ernovae C and K illustrate this

p ossible dierence where the b est t to C shows a frequency de

p endent maximum luminosity while for K the maxima at and

GHz app ear to b e the same

For our arguments on cosmic rays it is not relevant whether the

predecessor stars were Wolf Rayet stars or other massive stars with ex

tended winds p ermeated by strong magnetic elds We exp ect from the

similarity in the internal structure of the stars on the upp er main se

quence that their global prop erties such as the magnetic eld strength

generated should not b e drastically dierent Here we assume that we

can use the implied prop erties for Wolf Rayet stars just as for stars of

slightly lower mass which explo de into slow winds

V Electrons

A The injection of relativistic electrons

Ab ove we argued on the basis of a comparison of Wolf Rayet stars

and radio sup ernovae that there app ears b e a critical AlfvenicMach

number for the injection of electrons This critical Machnumber can

not b e pinp ointed to one sp ecic number at this time but app ears to

b e in the range of values of to

Levinson has argued on theoretical grounds that in

deed there is such a critical AlfvenicMachnumber for electron injection

his prediction contains a free parameter so that only consistency with

our argument can b e veried

Feldman privcomm at the Tucson meeting noted that

data from solar wind sho cks also show that there is critical Alfvenic

Machnumber for electron injection see Edmiston Kennel Ken

nel et al for a p ossible physical argument

What we are lacking now is a generalized theory to give this re

sult maybe even with a denite numerical value as well as an ar

gument what determines or eliminates electron injection b elow this

critical AlfvenicMachnumber It is p ossible to interpret the radio data

of stars with this concept since it leads to a latitude restriction and

thereby to a decrease in the radio emission a quantitative check sug

gests that then the radio emission is conned to a fairly small p olar

region while energetic protons can interact over a large part of the

hemisphere Nath Biermann b Biermann Strom Falcke

B The maximum energy of relativistic electrons

For ISMSN the maximum energy for electrons accelerated is given

by ab out GeV using a low density interstellar medium and

Synchrotron losses versus net acceleration acceleration minus adiabatic

losses as the limiting factor It follows that the observed high energy

PL BIERMANN

electron tail cannot b e attributed to normal sup ernova explosions into

the interstellar medium

For windSN the maximum energy is given by the consideration

already used to derive the maximum emission frequency for nonthermal

radio emission ab ove eq The corresp onding maximum electron

energy is approximately given by

E r B U TeV

emax pc

where r is the radius where the stellar wind changes character due

pc

to the shell formed by interaction with the interstellar medium This

radius may b e of order a few parsec and so the exp ected maximum

energy for a few parsec and a sho ck sp eed of order kmsec is ab out

TeV This is consistent with the observations summarized by Wieb el

that detect energetic electrons up to a few TeV particle energies

We leave the problem of how to get such high energy electrons through

the Galaxy to us and the implied consequences for another o ccasion

We conclude for here that for electrons ab ove ab out GeV

windsup ernovae are required to explain particle energies and sp ectrum

C The sp ectrum of relativistic electrons

At particle energies of order GeV and slightly higher the radio emis

sion of external galaxies is the most reliable indicator of this sp ectrum

Golla has discussed the b est data available accounting for all

the p ossible contribution from thermal radio emission and nds that

all excellent data a sample of seven galaxies with a well determined

nonthermal radiosp ectrum with an error less than in the sp ectral

index are compatible with a single sp ectral index of a p owerlaw energy

distribution for the electrons in this energy range of This

is very nearly the same sp ectral index as found for Hydrogen in cosmic

rays

Wieb el compiled all the data available in the literature



and nds ab ove ab out GeV a sp ectrum of E which is



to b e compared with E as the exp ected sp ectrum from

wind sho cks of steep er by unity than the injected sp ectrum due to

synchrotron losses

The particle energy of the switch b etween the two source sites and

the maximal electron energy observed as well as the p ositron fraction

remain to b e discussed in detail

D The protonelectron ratio in cosmic rays

During the expansion the energy of any individual relativistic elec

tron decreases by adiabatic expansion as the ratio of the radii from the

time when electron injection ceases to the time of when proton injection

ceases After this p oint the energy densities of b oth particles decrease

STELLAR WINDS SNe and CRs

together The simple dilution of the electron p opulation is paralled by

the steadily decreasing injection rate of protons since b oth the volume

and the ram pressure of the sho ck go down as radius to the p ower

during the adiabatic phase see Shklovsky eq should the

injection of protons cease at a later stage then this dierential dilu

tion has also to b e reckoned The energy density ratio of the electron

p opulation relative to that of the protons is given by p

p

R R

cr itp cr ite

This intermediate switch from electron injection with steady accel

eration to a simple adiabatic loss regime determines the net scaling of

the p ower of the electron p opulation to that of the proton p opulation

in the cosmic rays The observations suggest that from GeV the en

ergetic electron density is only ab out one p ercent eg Wieb el of

the density of the protons From this observed ratio the energy density

ratio integrated over the entire relativistic part of the particle sp ec

trum protons relative to electrons for the sp ectral index of is

given by In the standard leaky b ox mo del the ratio of energy den

sities of protons and electrons is not inuenced by propagation eects

This suggests an expansion of a factor of order in radius b etween the

time when electron injection ceases and the time when proton injec

tion ceases here we assume that electrons and protons originally have

comparable energy densities of their relativistic particle p opulations

We may have thus identied the origin of the observed electronproton

ratio in cosmic rays

E The shell thickness of Sup ernova remnants in the ISM

One clear prediction of the concept of fast convective turbulence

in the sho ck region is the fairly large thickness of the shell this shell is

the thickness of all the matter snowplowed together downstream and

in addition the length scale with the same column density upstream

Thus what we refer to as upstream in our mo del is fully contained in

the emission shell observed the average sho ck lo cation is deep inside

the emission shell The outer edge of the observed emission then in

this picture is the lo cation of the presently lo cally protuding sho ck

seen from the side Therefore the shell thickness in this mo del is

b oth upstream and downstream in the language used earlier and when

referred to the outer radius has the value

U U

r r

U U

for sup ernova explosions into the interstellar medium

This gives r r for a strong sho ck in a medium of constant

density It is imp ortant to note that in the concept discussed here this

PL BIERMANN

large thickness of the shell is traced to an instability of a cosmic ray

mediated sho ck front ie a sho ck strongly inuenced in its structure

by cosmic ray protons and other nuclei As a consequence a sho ck

front which do es not inject new particles into the system but just

squeezes the existing energetic particle p opulation will not show this

eect and should therefore b e much thinner in the simple limit of a

strong sho ck r r This is indeed what is consistent with the

Cygnus lo op Raymond priv comm however one problem

with the Cygnus lo op is the op en p ossibility that it is a warped sheet

which may app ear broader than it is really is

The data for some sources can b e read o published graphs Pye

et al Dickel et al Seward et al Dickel et al

and are close to this value for parts of the sources Kepler

SN Tycho and RCW Apparently the new radio emission of

sup ernova A also app ears to t this prediction StaveleySmith et

al A more detailed analysis of such data is clearly required

it is obvious that many sup ernova remnants are not nicely circularly

symmetric nor show a clear shell structure After all we know the

interstellar medium to b e extremely inhomogeneous The data are also

not always in agreement b etween radio and Xrays which after all

trace nonthermal particles and thermal hot gas when co oling b ecomes

imp ortant then the simple shell argument may not b e suciently ac

curate since it assumes constant density throughout the shell

The analoguous argument for winds remains to b e done and may

b e necessary to interpret the radio data for the nova GK Per

VI Airshower data other checks and consequences

We have discussed and reviewed the tests with airshower data else

where and it suces to summarize here the predictions and tests



We predict for protons a sp ectrum of E pap er CR I I I

the Akeno data t gives E pap er CR IV Radio data of normal



galaxies give E Golla



We predict for Helium and heavier elements E b e

low the knee the Akeno data also here give a sp ectrum very close to

prediction of E



We predict for the nuclei b eyond the knee E The

Akeno data give E The world data set of all go o d high energy

data also gives this sp ectrum as well as the cuto at the predicted par

ticle energy pap er UHE CR I I The Flys Eye data Bird et al

demonstrate that the chemical comp osition switches rapidly from a

heavy comp osition to a light comp osition near eV as predicted

already in Biermann d and in a brief form earlier by Bier mann Strittmatter

STELLAR WINDS SNe and CRs

We predict the particle energies at the b end or knee and the en

ergies of the various cutos the test with the Akeno data gives tted

values for these numbers close to prediction From the t the numeri

cal values are rather strongly constrained to within since we t

b oth vertical and slanted showers simultaneously in their showersize

distribution

It is obvious that we have diculty estimating the systematic error

resulting from the quite general use of limiting arguments such as

always strong sho cks always maximal curvature in the elements of the

fast convection always ignoring p ossible enhancements of the magnetic

eld strength imp ortant for the drifts The predictions and ts which

app ear to agree generally quite well illustrate the p ossible renements

required within the context of the approximations made

Further checks are p ossible with i the data analysis of Seo et



al who give a sp ectrum of E for Hydrogen and



E for Helium very close to our predictions ii The analysis

of Freudenreich et al who have argued for some time that the

chemical comp osition near the knee b ecomes heavily enriched iii the

newest Flys Eye data Bird et al which give a sp ectrum of



E b eyond the knee in the energy range eV to eV



for the monoo cular data and E in the energy range eV

to eV for stereo data while the classical data from Haverah Park

Cunningham et al also see Sun et al give a sp ectrum



of E b elow eV and the iv JACEE data presented at

Calgary Asakimori et al which also give sp ectra for Hydrogen

 

and Helium at GeV particle energies of E and E

resp ectively

Recently we have discussed all the low energy data for these sp ec

tra Biermann Gaisser Stanev and have shown that also the

overall normalization b etween ISMSN and windSN is close to that

exp ected on the basis of which stars have strong winds on the main

sequence and which do not this gives an overall ratio of energy con

tained in the two p opulations of to There we also attribute the

underabundance of Hydrogen and Helium to a the two dierent source

sites and b the enrichment in the winds of evolved massive stars see

also Silb erb erg et al

Also in other recent work we have shown that the low energy cos

mic rays may reionize the intergalactic medium after leaving normal

galaxies in galactic winds Nath Biermann and we have used

the cosmic ray ionization rate implied by molecular cloud data to esti

mate the lower energy cuto of the galactic cosmic rays to b etween

and MeV kinetic energy Nath Biermann

Further more we have used the concept of particle acceleration in

sho cks running through stellar winds and then hitting the surrounding

molecular shells to prop ose an explanation for the strong ray lines

PL BIERMANN

observed by COMPTEL in the Orion star forming region Blo emen et

al Nath Biermann b for comp eting mo dels see Bykov

Blo emen and Ramaty et al We consider this an imp ortant

test of the entire picture since we obtain a satisfactory match at once

for the nonthermal radio emission of early type stars the strong ray

line emission and the very low ray continuum emission This success

ful match required a consideration of the latitudedep endent particle

acceleration of a sho ck running through a stellar wind

We conclude that the mo del has passed a large number of tests

quite successfully allowing rst quantitative checks to b e made

However many questions remain to b e answered esp ecially as re

gards the transp ort of cosmic rays and the secondary to primary ratio

VI I Caveats

With our basic p ostulate of the the smal lest dominant scale we

have made a giant leap of faith in treating convection and turbulence

in an ionized magnetic medium This is the most glaring step in our

argument which it may take a long time to verify

Similarly in our derivation we used approximations from cosmic

ray transp ort theory far b eyond its proven range of validity The formal

agreement with the derivation of Drury gives reason to hop e that

we may not b e to o far from a prop er description

The agreement claimed b etween prediction and measurement of

cosmic ray sp ectra critically dep ends on the correction for galactic

transp ort here derived from a Kolmogorov sp ectrum The secondary

to primary ratio in cosmic ray nuclei as a function of energy suggests

clearly otherwise We have not demonstrated that the structure of the

interstellar medium and its temp oral b ehaviour really allows the sec

ondary to primary ratio to b e understo o d in the context of our theory

The data give conicting evidence as to the chemical comp osition

of the cosmic rays across the knee whether it is dominantly light or

increasingly heavy We just show that the Akeno airshower data are

consistent with the second p ossibility However it can b e exp ected that

MACROEASTOP data will shed considerable light on this issue

For intergalactic transp ort of cosmic rays we have used an approx

imation of nearly straight line paths a connection with the bubble

structure of the galaxy distribution may exist A comparison with the

existing data base of high energy events is still outstanding

For massive stars we have suggested a theory for the winds based

on somewhat stronger magnetic elds than hitherto used and have

emphasized the p ossibility of a more tangential magnetic eld geometry

near the surface of the star However we have not actually calculated

the structure of such a wind nor have we b een able to compare it with observations to the detail desirable

STELLAR WINDS SNe and CRs

While the apparent agreement of the predictions with data gives

hop e that it is worth pursueing the development of the theory a lot of

work remains to b e done

VI I I Summary

Here we concentrate on the consequences for stars the implications

for cosmic rays have b een describ ed elsewhere Biermann b c d

the preceding section also gives the more imp ortant implications and

further questions

Novae OB and Wolf Rayet stars show evidence for particle

acceleration in winds the theory prop osed can account for what is

known of the sp ectra luminosities and temp oral b ehaviour Better

and more complete data are needed

White dwarfs provide a check on the notion that a magnetic

dynamo op erates inside the inner convection zone in massive stars this

dynamo may provide the fairly strong magnetic eld which we argue

is present in the stellar winds of massive stars

Wolf Rayet stars may derive partially the momentum in their

wind from the pressure gradient of the tangential magnetic eld their

pressure gradient acts as an amplier on a wind which has some ini

tial acceleration most likely from linedriving p ossibly in a multiple

scattering mo de

The comparison of the various stellar radio sources leads to the

concept of a critical AlfvenicMachnumber for electron injection this

concept may b e the basis for understanding of the protonelectron ratio

in the observed galactic cosmic rays

Sup ernova sho cks in the stellar winds may provide the sources

of galactic cosmic rays in a Helium and heavier elements b electrons

b eyond ab out GeV c all the way across the knee to ab out

GeV where the chemical comp osition is mostly heavy We have b een

able to quantitatively test the theory using airshower data and other

recent cosmic ray data

Various alternative mo dels exist

A p ostulated galactic wind mo del Jokipii Morll may

accelerate particles at a galactic wind termination sho ck this is

argued to contribute particles over the entire range of particle en

ergies from low energies to the end of the cosmic ray sp ectrum

At low energies the distance which particles can reach upstream

from the sho ck when they stream back to our Galaxy is limited by

V where V refers to the galactic wind velocity and is the

W W

diusion co ecient for energetic particles at the relevant energies

This distance is so small for any reasonable value for the diu

sion co ecient that low energy particles cannot reach the Galaxy This requires that we have two dierent source p opulations which

PL BIERMANN

merge near the knee which in turn implies considerable netuning

of the source parameters At the high particle energies it is very

dicult to see that the particles can b e contained in the Galaxy

Berezinskii et al IVx

The multiple sho cks in the environment of OB sup erbubbles and

young sup ernova remnants Bykov Toptygin Polcaro

et al Bykov Fleishman Ip Axford

may also contribute

The cosmic background of active galactic nuclei may contribute

through the pro duction of energetic neutrons which convert back

to protons Prothero e Szab o

For all such mo dels a clear prediction of the sp ectrum and its chemical

abundance distribution as well as a detailed check with the airshower

size distribution b oth for slanted and oblique showers is desirable

We emphasize that our prop osal as far as the galactic cosmic rays

are concerned rests on a plausible but nevertheless sp eculative assump

tion ab out the nature of the transp ort of energetic particles in p erp en

dicular sho ck waves namely that there are large fast convective motions

across the average lo cation of the sho ck interface this notion is how

ever supp orted by radio p olarization observations of young sup ernova

remnants as well as theoretical arguments as briey describ ed ab ove

The mo del has predictive power We have given predictions and checks

ab ove However a large amount of work remains to b e done

There are many obvious tasks to b e done next as we have given a

number of imp ortant steps and checks for the cosmic ray asp ect else

where we concentrate here on the ramications concerning stars and

stellar evolution

Direct observational tests for the strength of the magnetic elds

in OB stars Wolf Rayet stars red sup ergiant winds and radio

novae are critically imp ortant The mo del presented ab ove un

equivocally dep ends on the magnetic eld strengths prop osed One

p ossibility to determine the strength of the magnetic eld in winds

app ears to b e the time dep endent dep olarization of the radio emis

sion from a pulsar orbiting an upp er main sequence star such as

B Thorsett rep orting on radio observations made by

RN Manchester et al

The radio sp ectral evolution of novae single OB and WR stars as

well as radio sup ernovae including the latitude distribution and

radiative transfer should b e calculated to make the predictions

more accurate and thus more testable

The dynamo pro ducing the magnetic eld in the inner convective

zone in upp er main sequence stars needs to b e mo delled as well as

the transp ort of magnetic ux through the radiative zone outwards

in order to see whether the magnetic elds and their geometry prop osed can really b e generated and transp orted

STELLAR WINDS SNe and CRs

We need a detailed mo del of a stellar wind which starts with

maybe line driving in the multiple scattering mo de and then re

ceives additional outward momentum from the gradient of the tan

gential magnetic eld

Finally we need a prop er stellar evolution calculation taking into

account up to maximal rotation and up to maximum magnetic

elds which may play an imp ortant role in the nal stages of stellar

evolution

Acknowledgements

First of all I wish to thank my graduate student H Seemann for

checking all equations in this article and going over it with a nely

to othed comb Second I have to thank my collab orators in the on

going work partially rep orted here Drs JP Cassinelli H Falcke

TK Gaisser JR Jokipii H Meyer BB Nath J Rachen ES Seo

T Stanev RG Strom and B Wieb el Further evolution of these

ideas was prompted by intense discussions and exchanges again with

my collab orators as well as Drs G Auriemma VS Berezinsky P

Bhattacharjee JH Bieging A Bykov P Charb onneau J Cronin L

Dedenko JR Dickel R Diehl V Dogiel J Eilek P Evenson M

Giller F Giovannelli C Jarlskog J Kirk G Mann JF McKenzie

M Nagano SP Owocki MI Pravdin VS Ptuskin R Ratkiewicz

J Raymond ER Seaquist FD Seward VV Usov G Webb J

Wdowczyk and G Zank Helpful comments on various parts of this

manuscript were received from VS Berezinsky P Charb onneau JR

Dickel RN Manchester JF McKenzie SP Owocki ER Seaquist

P Song J Raymond FD Seward and GP Zank as well as two un

known referees I noted the contribution from the interaction at various

meetings ab ove I am grateful for the organizers for inviting me to their

conferences always in b eautiful settings such as the island Vulcano

Imp ortant help in nding some of the older references was received

from Dr J Peiderer and the librarian of the MPI for Fluid Mechan

ics High Energy Physics with the author is supp orted by a NATO travel grant CRG

PL BIERMANN

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