Accelera'on  and  propaga'on  of  cosmic  rays  -­‐  II  

2/7/12   Tom  Gaisser   2  

Energe'cs  of  cosmic  rays  •  Total  local  energy  density:    

–  (4π/c)  ∫  Eφ(E)  dE            ~  10-­‐12  erg/cm3  ~  B2  /  8π  –  B  ~  3  µGauss  in  ISM  

•  Comparable  energy  density  in  magne'c  fields  and  CR      –  consistent  with  model  of  CR  

diffusing  in  the  Galaxy  •  Milky  Way  Galaxy  

–  Spiral  Galaxy  –  Disk  with  radius  ~15  kpc  –  Thickness  0.5  kpc  –  Bulge  around  galac'c  center  –  1  pc  =  3  x  1018  cm  –  Vdisk  ~  1067  cm3  

Spectral Energy Distribution (linear plot shows most E < 100 GeV) (4π/c) Eφ(E) = local differential CR energy density

14.1 The Galaxy 137

Figure 14.1 Artistic view of the Galaxy from above and from the side.REF: Benjamin, R. A. (2008). ”The Spiral Structure of the Galaxy: Some-thing Old, Something New...”. In Beuther, H.; Linz, H.; Henning, T.(ed.). Massive Star Formation: Observations Confront Theory. 387. As-tronomical Society of the Pacific Conference Series. pp. 375. Bibcode2008ASPC..387..375B. Lay summary (2008-06-03). See also Bryner, Jeanna(2008-06-03). ”New Images: Milky Way Loses Two Arms”. Space.com. Re-trieved 2008-06-04.

the galactic disk at about 8.5 kpc from the galactic center on the inner edgeof the Orion-Cygnus Arm.

The majority of standard matter (to be distinguished from dark matter)is concentrated in the thin disk and it is composed by stars of various agesand interstellar matter (ISM). The ISM is filled by gas, dust and by cosmicrays and it accounts for 10-15 % of the total mass of the galactic plane. It isvery inhomogeneously distributed at small scales and it is mostly confinedto discrete clouds. Only few percent of the interstellar volume is occupiedby dense accumulation of ISM. For understanding the origin of galactic cos-mic rays and the energy involved, we review here the ISM and star formingregions. Of particular interest is the feedback exercised by supernovae explo-sions and their remnants on the ISM and related trigger of star formation.Moreover, we describe the galactic center region and for completeness thedark matter halo.

Ar'st’s  view  of  Milky  Way  

Power  requirement    •  Power  needed:  

–  (4π/c)  ∫  Eφ(E)  /  τesc(E)  dE  –   τesc  ~  107  E-­‐0.33  yrs  –  φ(E)  ~  1.7  x  E-­‐2.7  (GeVcm2  sr  s)-­‐1  –  Evaluate  integral:  ~  10-­‐26  erg/cm3/s  –  (use  627  GeV  /  erg)  – Mul'ply  by  Vdisk  ~  1067  cm3    and  3  x  107  s/yr  –  3  x  1048  erg/yr  

•  Supernova  power:  –  1051  erg/SN  –  3  SN  /  century  –  3  x  1049  erg  /  yr  

•  Need  10%  efficiency  to  accelerate  CR  with  SNR  

Stellar  collapse  à  supernova  

Energy  of  SN  ejecta:  

 for  

2/7/12   Tom  Gaisser   6  

Cosmic  rays  in  the  Galaxy  •  Supernova  explosions  energize  the  

ISM  –  ~1%  Kine'c  energy;                                                  neutrinos  ~  99%  

–   >10%  of  kine'c  energy  à  CR  accelera'on  

–  Energy  density  in  CR  ~  B2/8π –  SN  &  CR  ac'vity  drives  Galac'c  wind  

into  halo  (Parker)  

–  CR  diffuse  in  larger  volume  

–  Eventually  escape  Galaxy  

April  9,  2009     Tom  Gaisser   7  

Supernova progenitor

SN ejecta Shocked ISM

Supernova  blast  wave  accelera'on  

Unshocked ISM

SNR expands into ISM with velocity ~ 104 km/s. Drives forward shock

Forward shock

u1

u1

Particle with E1

E2 = ξ E1

Contact discontinuity

TSN ~ 1000 yrs before slowdown Emax ~ Z x 100 TeV

Picture  in  upstream  frame  

Expanding  SN  ejecta  

Nearly  plane  shock  moving  out  ahead  of  “piston”  with  velocity  -­‐u1      (u1>  sound  speed)  

upstream  

downstream  

Picture  in  shock  frame  

u1  

u2  

 u1  –  u2  =  V  

upstream  

downstream  

V  =  speed  of  shocked,  turbulent  gas  

 u1  /  u2    =    4    for  a  strong  shock  

Astroteilchenphysik  2009   Tom  Gaisser  Cosmic  rays  -­‐  1   10  

Accelera'on  

Astroteilchenphysik  2009   Tom  Gaisser  Cosmic  rays  -­‐  1   11  

April  9,  2009     Tom  Gaisser   12  

Diffusion  by  irregulari'es  in  B  Cosmic-Ray Modulation Equations

Fig. 1 Charged particle motion in a magnetic field. (a) In a uniform magnetic field the particle has a spiralorbit with a gyroradius rg = P/Bc. (b) When the field is non-uniform the particle drifts away from a fieldline due to the gradient and curvature of the field. (c) When a particle meets a kink in the field that has a scalelength ! rg , all particles will progress through the kink (but they may drift to adjacent field lines while doingso). (d) Likewise, if rg ! scale size of the kink, all particles will pass through it without being affected much.(e, f, g) When rg ≈ scale size of the kink, it depends on the gyrophase of the motion when the particles startsto feel the kink whether it will go through the kink (e), be reflected back (f), or effectively get stuck in thekink (g). This process is called pitch-angle scattering along the field. (h) When particles meet such a kink,there is also a scattering in phase angle, which leads to scattering across the field lines, but such that κ⊥ $ κ‖

a circle with radius rg = mv/(qB) = p/(qB). This implies that the gyroradius depends ontwo particle properties, namely its momentum and charge. For this reason we introduce theconcept of rigidity, defined as P = p/q . Then rg = P/B , which says that the gyroradiusdepends on only one particle property and on the field strength.

The SI-units of rigidity are kg m s−1 C−1 or J s m−1 C−1, and this is cumbersome to use.It can be translated into the much more useful unit of Volt (V) by noting from (1) that pchas the same units as E. Thus, if one rather defines rigidity as P = pc/q , it has dimensionsof energy per unit charge, or potential. If energy (and pc) is expressed in eV, and charge interms of the number Z of elementary charges, i.e. q = Ze where e = 1.602 × 10−19 C, thenP has units of Volt (V). Thus, the formal definition of rigidity is

P ≡ pc/(Ze),

with the gyroradius given by rg = P/Bc.Putting this into (1) gives the relationship P = (A/Ze)2√T (T + 2E0) between the rigid-

ity of a particle and the kinetic energy per nucleon of that particle. Bearing in mind thatm = m0/

√1 − β2, one gets the universal relationship

P = pc/(Ze) = (A/Ze)√

T (T + 2E0) = (A/Ze)β(T + E0), (3)

“Scalering”  occurs  most  effec'vely  when  size  of  irregularity  ~  gyroradius  

Space Sci RevDOI 10.1007/s11214-011-9819-3

Cosmic-Ray Modulation Equations

H. Moraal

Received: 6 December 2010 / Accepted: 5 August 2011© Springer Science+Business Media B.V. 2011

Abstract The temporal variation of the cosmic-ray intensity in the heliosphere is calledcosmic-ray modulation. The main periodicity is the response to the 11-year solar activitycycle. Other variations include a 27-day solar rotation variation, a diurnal variation, and ir-regular variations such as Forbush decreases. General awareness of the importance of thiscosmic-ray modulation has greatly increased in the last two decades, mainly in communitiesstudying cosmogenic nuclides, upper atmospheric physics and climate, helio-climatology,and space weather, where corrections need to be made for these modulation effects. Pa-rameterized descriptions of the modulation are even used in archeology and in planning theflight paths of commercial passenger jets.

The qualitative, physical part of the modulation is generally well-understood in thesecommunities. The mathematical formalism that is most often used to quantify it is the so-called Force-Field approach, but the origins of this approach are somewhat obscure and itis not always used correct. This is mainly because the theory was developed over more than40 years, and all its aspects are not collated in a single document.

This paper contains a formal mathematical description intended for these wider commu-nities. It consists of four parts: (1) a description of the relations between four indicators of“energy”, namely energy, speed, momentum and rigidity, (2) the various ways of how tocount particles, (3) the description of particle motion with transport equations, and (4) thesolution of such equations, and what these solutions mean. Part (4) was previously describedin Caballero-Lopez and Moraal (J. Geophys. Res, 109: A05105, doi:10.1029/2003JA0103582004). Therefore, the details are not all repeated here.

The style of this paper is not to be rigorous. It rather tries to capture the relevant tools todo modulation studies, to show how seemingly unrelated results are, in fact, related to oneanother, and to point out the historical context of some of the results. The paper adds no newknowledge. The summary contains advice on how to use the theory most effectively.

Keywords Cosmic rays · Modulation · Force field · Transport equation

H. Moraal (!)Space Research Centre, North-West University, Potchefstroom 2520, South Africae-mail: harm.moraal@nwu.ac.za

2/28/12   Gaisser   13  

April  9,  2009     Tom  Gaisser   14  

Next  step  is  to  average  over  cos  θ2    and  cos  θ1  

April  9,  2009     Tom  Gaisser   15  

Distribu'on  of  exit  angles  (cos  θ2  )  

cos  θ2  averages  to  0  so  

Distribu'on  of  entrance  angles  (cos  θ1  ):  

 cos  θ1  averages  to    -­‐V  /  3c    so  

cos  θ2  averages  to  2/3  so  that  

cos  θ1  averages  to  -­‐2  /  3  so  that  

April  9,  2009     Tom  Gaisser   16  

April  9,  2009     Tom  Gaisser   17  

April  9,  2009     Tom  Gaisser   18  

Other  expressions  for  Emax  

Rewrite  as  

Interpreta'on:  Gyroradius  of  par'cle  must  be  <  size  of  accelerator  

Can  also  relate  to  power  needed  in  source