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Effect of re-acceleration on cosmic ray components
Adv. Space Res. Vol. 9. No. 12. pp. (12)93—(12)96. 1989 Printed in Great Britain. All rights reserved.
0273—1177/89 S0.00 + .50 Copyright © 1989 COSPAR
EFFECT OF RE-ACCELERATION ON COSMIC RAY COMPONENTS S. A. Stephens* and R. L. Golden Particle Astrophysics Laboratory, New Mexico State University, Box 3—0, Las Cruces, NM 88003, U.S.A.
ABSTRACT Re-acceleration of cosmic rays in interstellar space has been studied In detail in order to examine the behaviour of the ratios of secondary to primary nuclei in cosmic radiation. It Is found that modest acceleration In a confinement region, where particles escape more freely at high energies, provides a better fit to the observed data. We have studied the effect of reacceleration on the spectral shape of proton and helium components of cosmic rays. We have examined two different models and showed that re-acceleration provides a poor fit to the observed proton data. I NTRODUCTION If supernova remnants fill a large fraction of the hot tenuous phase of the interstellar medium, it Is expected that the shock waves from the overlapping remnants could accelerate the ambient cosmic rays in the Galaxy /1/. Strong shocks would modify the spectral shape considerably, unlike the weak shocks /2/. Since the spectral shape of secondary cosmic ray nuclei is steeper than the primary nuclei, it is expected that strong shocks do not play any significant role in the interstellar medium. Continuous acceleration of cosmic rays Is also questioned on the basis of the observed variation of the ratio of secondary to primary nuclei with energy /3/. However, cosmic rays may frequently encounter weak shocks and acquire a small energy increment every time. It is also recognized that if cosmic ray scattering in the interstellar medium is due to the magneto hydrodynamic turbulence, the same scattering may lead to statistical particle acceleration /4/. Because of these, many attempts have been made to study the effect of re-accieration of cosmic rays in the Galaxy /5-9/. It is believed that cosmic ray confinement In the Galaxy Is energy dependent and that higher energy particles escape more freely than the low energy ones. It is then expected that re-acceleration could modify the spectral shape of cosmic rays in the energy region, where they spent more time in the Galaxy. In the case of heavier nuclei, the interaction loss dominates and the effect may not be visible. However, for the lighter components of cosmic rays of charge < 2, this effect would be recognizable. In this paper, we have attempted to calculate the spectral shape of proton and helium nuclei in the interstellar space, using models which Incorporate re-acceleration. We then subject these spectra to solar modulation in order to derive their spectra near the Earth and compare the results with the available data. METHOD OF CALCULATION We have made use of two models to calculate the spectra of protons and helium nuclei in the cosmic rays. The first one is due to Wandel et al /7/, who formulated a transport equation for re-distributed acceleration and explained many heavy nuclei observations. One can write this as,
*
On leave from Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India.
(12)93
(12)94
S. A. Stephens and R. L. Golden
dJ(E)/dt
a{J(E)dE/dt}/~E - J(E){nv(E)~(E)
+
l/Tes
+
l/Tac}
(1l)RT(E){JJ(E~)R~’~(E)dE}/Tac+ Q(E) (1) E 0 Here, the LHS gives the rate of change of flux at energy E, n the number density of hydrogen, v the velocity, R the rigidity, Tes and Tac are the escape and shock encounter tImes respectively, c is the cross-sectIon for interaction and 7 is the shock acceleration Index. In this equation, the 1st term on the RHS Is the continuous energy loss term and we consider here the ionization loss. 2nd term is the loss of particle at energy E due to interaction, escape and shock encounter. According to Letaw et al /10/, the escape time is assumed to be of the form, 6 (R 6 years (2) Tes 6.3xl0 0/R) for a density of 0.5 hydrogen atom/cc, where 6=0.6, and Tac 7.6x106 years. The value of R 0 corresponds to a kinetic energy of 1 GeV/n for helium nuclei and 2 GeV for protons. The 3rd term is the acceleration term, where the value of 1=4.0 and the Integration limit E0 is determined by the ionization loss. The last term is the production term which is taken to be a power law in rigidity with a spectral index of -(2.75-6). We have included In this term the contribution from heavier elements in an approximate manner. +
Equation (1) is solved simultaneously for protons and helium nuclei, as the transport equation for protons contains a production term due to spallation of helium nuclei. We have used Runga-Kutta step wise integration technique and the integration is continued till the term dJ(E)/dt —~.0. The calculated equilibrium spectra in the Galaxy are then subjected to solar modulation using one dimensional force-field analytic approximation /11,12/. Thus, we derive the energy spectra of protons and helium nuclei near the Earth during solar minimum and maximum periods. In the second model, Osborne and Ptuskin (0 & P) /4/ considered continuous acceleration due to the magneto hydrodynamic waves. They showed that the effect of accieration is equivalent to modifying the confinement time of cosmic rays in the Galaxy and they obtained an expression for the effective life time as, 6 Cl.+k(R 6} (3) Teff A (R0/R) 0/R)2 From a comparison with the observed ratio of secondary to primary nuclei as a function of energy, they derived A 4.2, 6 =0.33, k=1 and R 0 = 5.5 GV. Since the effective life time is a combination of Tes and Tac, as l/Teff l/Tes + l/Tac, one can deduce the effective acceleration time from Equation (3). Taking Tes = A (R0/R)~, the effective acceleration time becomes 6 (4) Tac = Tes - A(R0/R) where the negative sign indicates acceleration. These expressions have been used in the 2nd term in the Equation (1). For the production term, we have used a power law spectrum in rigidity with a spectral index of -(2.75-6 ). The Integration is carried out simultaneously as described earlier without the acceleration term contained in the integral. The resultant equilibrium spectra of protons and helium nuclei are then subjected to solar modulation as discussed earlier. RESULTS We have plotted in Figure 1 the equilibrium spectra of cosmic ray protons in interstellar space by Curves marked A. The solid curve correspond to Wandel et al model, while the dashed curve is for 0 & P model. These two spectra differ very much at low energies. Curves marked B in this figure are the spectra near the Earth during the period of minimum solar modulation and the curves marked C are for the period of maximum solar modulation. The difference between these models persists with lesser magnitude as the modulation increases. The curve drawn with circles is the observed proton spectrum during the solar minimum period. This is the fitted spectrum to the observed flux values /13/. It may be pointed out that the data below about 5 GeV is very meagre. The predicted spectra for the solar minimum period (Curves marked B) are well above the data. Thus the re-accleratlon predicts too large a flux of protons at low energies. One may be able to reconcile the Wandel et al spectrum with the data by introducing stronger modulation.
Re-Acceleration of Cosmic Ray Components
However, it will be seen that such a procedure will make the helium spectrum inconsistent with the data. The calculated spectra of helium nudel In the Galaxy are shown in FIgure 2 by curves marked A for these two models. The curves marked B and C are respectively for the solar minimum and maximum periods. The data points represent the fitted curve to the observed data /12/. This spectrum is well determined down to about 2 GeV/n. It can be seen that the predicted spectrum using the Wandel et al model seem to agree with the data, while that due to 0 & P is not consistent with the data. Comparing the proton and helium spectra, one may conclude cos— mic rays that in re-acceleration the interstellar ofmedium does not seem to reproduce the observed spectral shape of light cosmic ray nuclei In a consistent manner.
(12)95
_________________________
‘‘“I
‘‘‘“I
/ UI ~
~..,
1O~
,~
-
~
~
1/’
~
\
~
-~
10~
~
,“
-
‘s
z H
N
2-
~ ~o
A. INTERSTELLAR Z
B. SOLAR MINIMUM C. SOL?.R MAXIMUM
DISCUSSION 101
From the above study of the spectra ~ of proton and helium nuclei in the ~ ... Galaxy, it appears that the accelera1. 0 • tion mechanism suggested by Osborne 10 10 10 and Ptuskin over estimates the flux ENERGY PER NUCLEON ( Gay of protons and helium nuclei. One may rig. 1. Calculated proton energy spectrum bear in mind that the simplified ap— the are shown. Solid and dashed curves proach made here to derive pro ton and helium spectra using this model, may not be valid at very low energies. However, above a few GeV/n this model predicts heavy nuclei abundance satisfactorily. In this energy region, e a comparison of the predicted spectrum with the observed data shows ~ that this model predicts more proton 1 flux than that observed by a factor c’i of two even at about 5 GeV, where the .~. data is reliable. Similarly, the pre— -...~ dicted helium spectrum during the ~ ., solar minimum period does not also ‘~ f it the data. ~ l0~ ~ In the case of Wandel et al model,the predicted proton spectrum does not fit the proton data, though the calculated spectrum reproduces the He data. As we have pointed out that if one makes adjustment with the modulation parameters so as to make the reacclerated spectrum consistent with the proton data, the same modulation parameters would predict too low a flux of helium nuclei. Therefore, we
2 10
near are
~‘
~ Nb1
~
~,
~ E~ 1~o -
conclude at this stage that reacceleration of cosmic rays in inter— stellar space would be inconsistent with the observed spectra of proton and helium nuclei. It may be seen from the parameters used by these authors for the escape time from the confinement volume,that low energy particles spent most of their time in the Galaxy. If one assumes that the energy dependent con-
~ c.
~
SO1J~MAXIMUM
~ io—l
100
101
ENERGY PER NUCLEON
io2
C GeV
Fic~. 2. Helium energy spectrum near the Earth based on re-acceleration models are shown by solid and dashed curves. Curve marked with
circles
represent
the observed spectrum.
(12)96
S. A. Stephens and R. L. Golden
finement Is apllicable only above certain rigidity R 0, it is possible to suppress the enhanced flux below about a few GeV. However, this procedure may not affect the spectral shape above about 10 GeV. Our calculations show that the resultant spectral shape of the proton spectrum in the energy region between 10 and 100 GeV is steeper by an index of about 0.1 compared to that of helium nuclei. One may be able to test this small difference with the future experiments. . Acknowledgement. This work is supported by NASA grant NGWA 110. REFERENCES -
~1.,237, 793 (1980).
1.
R.D. Blanford and J.P. Ostriker, Astrophys.
2.
H. Moraal and W.I. Axford, Astron. Astrophys., 125, 204 (1983).
3.
R. Cowsik, Astrophys.
4.
J.L. Osborne and V.S. Ptuskin, Proc. (1987).
5.
M. Simon, 14. Heinrich and K.D. Mathais, Astrophys.
6.
M. Simon, U. Heinbach and Ch. Koch, Astrophys. .1., 320, 699 (1987).
7.
A. Wandel, D. Eichler, J.R. Letaw, Astrophys. L. 316, 676 (1987).
8.
M. Giler, B. Szabelski, J. Wodwczyk and A.W. Wolfendale, Proc. 2Q..tXI Cosmic ~y Conf., 2, 211 (1987).
.In~L
9.
Ph. Ferrando and A. Soutoul, Proc. ~.Q~jjInt. Cosmic B~iConf., (1987)
2,
231
and
A.
L. 241, 1195 (1980). 2Q.~cJI
R.
~
Cosmic
~
Conf., 2, 218
L~ 300, 32 (1986).
Silberberg
and
C.S.
10. 3.R. Letaw, R. Sllberberg, C.H. Tsao, D. Elchler, M.M. Shapiro Wandel, Proc. 2.Q..th jj~ Cosmic ~y Conf., 2, 222 (1987).
Tsao,
11.. J.S. Perko, Astron. Astrophys., 184, 119 (1987). 12. S.A. Stephens and R.L. Golden, this issue. 13. W.R. Webber, R.L. Golden and S.A. Stephens, Proc. Conf., 1, 325 (1987).
2.Q.~Jj ~
Cosmic
R~
0273—1177/89 S0.00 + .50 Copyright © 1989 COSPAR
EFFECT OF RE-ACCELERATION ON COSMIC RAY COMPONENTS S. A. Stephens* and R. L. Golden Particle Astrophysics Laboratory, New Mexico State University, Box 3—0, Las Cruces, NM 88003, U.S.A.
ABSTRACT Re-acceleration of cosmic rays in interstellar space has been studied In detail in order to examine the behaviour of the ratios of secondary to primary nuclei in cosmic radiation. It Is found that modest acceleration In a confinement region, where particles escape more freely at high energies, provides a better fit to the observed data. We have studied the effect of reacceleration on the spectral shape of proton and helium components of cosmic rays. We have examined two different models and showed that re-acceleration provides a poor fit to the observed proton data. I NTRODUCTION If supernova remnants fill a large fraction of the hot tenuous phase of the interstellar medium, it Is expected that the shock waves from the overlapping remnants could accelerate the ambient cosmic rays in the Galaxy /1/. Strong shocks would modify the spectral shape considerably, unlike the weak shocks /2/. Since the spectral shape of secondary cosmic ray nuclei is steeper than the primary nuclei, it is expected that strong shocks do not play any significant role in the interstellar medium. Continuous acceleration of cosmic rays Is also questioned on the basis of the observed variation of the ratio of secondary to primary nuclei with energy /3/. However, cosmic rays may frequently encounter weak shocks and acquire a small energy increment every time. It is also recognized that if cosmic ray scattering in the interstellar medium is due to the magneto hydrodynamic turbulence, the same scattering may lead to statistical particle acceleration /4/. Because of these, many attempts have been made to study the effect of re-accieration of cosmic rays in the Galaxy /5-9/. It is believed that cosmic ray confinement In the Galaxy Is energy dependent and that higher energy particles escape more freely than the low energy ones. It is then expected that re-acceleration could modify the spectral shape of cosmic rays in the energy region, where they spent more time in the Galaxy. In the case of heavier nuclei, the interaction loss dominates and the effect may not be visible. However, for the lighter components of cosmic rays of charge < 2, this effect would be recognizable. In this paper, we have attempted to calculate the spectral shape of proton and helium nuclei in the interstellar space, using models which Incorporate re-acceleration. We then subject these spectra to solar modulation in order to derive their spectra near the Earth and compare the results with the available data. METHOD OF CALCULATION We have made use of two models to calculate the spectra of protons and helium nuclei in the cosmic rays. The first one is due to Wandel et al /7/, who formulated a transport equation for re-distributed acceleration and explained many heavy nuclei observations. One can write this as,
*
On leave from Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India.
(12)93
(12)94
S. A. Stephens and R. L. Golden
dJ(E)/dt
a{J(E)dE/dt}/~E - J(E){nv(E)~(E)
+
l/Tes
+
l/Tac}
(1l)RT(E){JJ(E~)R~’~(E)dE}/Tac+ Q(E) (1) E 0 Here, the LHS gives the rate of change of flux at energy E, n the number density of hydrogen, v the velocity, R the rigidity, Tes and Tac are the escape and shock encounter tImes respectively, c is the cross-sectIon for interaction and 7 is the shock acceleration Index. In this equation, the 1st term on the RHS Is the continuous energy loss term and we consider here the ionization loss. 2nd term is the loss of particle at energy E due to interaction, escape and shock encounter. According to Letaw et al /10/, the escape time is assumed to be of the form, 6 (R 6 years (2) Tes 6.3xl0 0/R) for a density of 0.5 hydrogen atom/cc, where 6=0.6, and Tac 7.6x106 years. The value of R 0 corresponds to a kinetic energy of 1 GeV/n for helium nuclei and 2 GeV for protons. The 3rd term is the acceleration term, where the value of 1=4.0 and the Integration limit E0 is determined by the ionization loss. The last term is the production term which is taken to be a power law in rigidity with a spectral index of -(2.75-6). We have included In this term the contribution from heavier elements in an approximate manner. +
Equation (1) is solved simultaneously for protons and helium nuclei, as the transport equation for protons contains a production term due to spallation of helium nuclei. We have used Runga-Kutta step wise integration technique and the integration is continued till the term dJ(E)/dt —~.0. The calculated equilibrium spectra in the Galaxy are then subjected to solar modulation using one dimensional force-field analytic approximation /11,12/. Thus, we derive the energy spectra of protons and helium nuclei near the Earth during solar minimum and maximum periods. In the second model, Osborne and Ptuskin (0 & P) /4/ considered continuous acceleration due to the magneto hydrodynamic waves. They showed that the effect of accieration is equivalent to modifying the confinement time of cosmic rays in the Galaxy and they obtained an expression for the effective life time as, 6 Cl.+k(R 6} (3) Teff A (R0/R) 0/R)2 From a comparison with the observed ratio of secondary to primary nuclei as a function of energy, they derived A 4.2, 6 =0.33, k=1 and R 0 = 5.5 GV. Since the effective life time is a combination of Tes and Tac, as l/Teff l/Tes + l/Tac, one can deduce the effective acceleration time from Equation (3). Taking Tes = A (R0/R)~, the effective acceleration time becomes 6 (4) Tac = Tes - A(R0/R) where the negative sign indicates acceleration. These expressions have been used in the 2nd term in the Equation (1). For the production term, we have used a power law spectrum in rigidity with a spectral index of -(2.75-6 ). The Integration is carried out simultaneously as described earlier without the acceleration term contained in the integral. The resultant equilibrium spectra of protons and helium nuclei are then subjected to solar modulation as discussed earlier. RESULTS We have plotted in Figure 1 the equilibrium spectra of cosmic ray protons in interstellar space by Curves marked A. The solid curve correspond to Wandel et al model, while the dashed curve is for 0 & P model. These two spectra differ very much at low energies. Curves marked B in this figure are the spectra near the Earth during the period of minimum solar modulation and the curves marked C are for the period of maximum solar modulation. The difference between these models persists with lesser magnitude as the modulation increases. The curve drawn with circles is the observed proton spectrum during the solar minimum period. This is the fitted spectrum to the observed flux values /13/. It may be pointed out that the data below about 5 GeV is very meagre. The predicted spectra for the solar minimum period (Curves marked B) are well above the data. Thus the re-accleratlon predicts too large a flux of protons at low energies. One may be able to reconcile the Wandel et al spectrum with the data by introducing stronger modulation.
Re-Acceleration of Cosmic Ray Components
However, it will be seen that such a procedure will make the helium spectrum inconsistent with the data. The calculated spectra of helium nudel In the Galaxy are shown in FIgure 2 by curves marked A for these two models. The curves marked B and C are respectively for the solar minimum and maximum periods. The data points represent the fitted curve to the observed data /12/. This spectrum is well determined down to about 2 GeV/n. It can be seen that the predicted spectrum using the Wandel et al model seem to agree with the data, while that due to 0 & P is not consistent with the data. Comparing the proton and helium spectra, one may conclude cos— mic rays that in re-acceleration the interstellar ofmedium does not seem to reproduce the observed spectral shape of light cosmic ray nuclei In a consistent manner.
(12)95
_________________________
‘‘“I
‘‘‘“I
/ UI ~
~..,
1O~
,~
-
~
~
1/’
~
\
~
-~
10~
~
,“
-
‘s
z H
N
2-
~ ~o
A. INTERSTELLAR Z
B. SOLAR MINIMUM C. SOL?.R MAXIMUM
DISCUSSION 101
From the above study of the spectra ~ of proton and helium nuclei in the ~ ... Galaxy, it appears that the accelera1. 0 • tion mechanism suggested by Osborne 10 10 10 and Ptuskin over estimates the flux ENERGY PER NUCLEON ( Gay of protons and helium nuclei. One may rig. 1. Calculated proton energy spectrum bear in mind that the simplified ap— the are shown. Solid and dashed curves proach made here to derive pro ton and helium spectra using this model, may not be valid at very low energies. However, above a few GeV/n this model predicts heavy nuclei abundance satisfactorily. In this energy region, e a comparison of the predicted spectrum with the observed data shows ~ that this model predicts more proton 1 flux than that observed by a factor c’i of two even at about 5 GeV, where the .~. data is reliable. Similarly, the pre— -...~ dicted helium spectrum during the ~ ., solar minimum period does not also ‘~ f it the data. ~ l0~ ~ In the case of Wandel et al model,the predicted proton spectrum does not fit the proton data, though the calculated spectrum reproduces the He data. As we have pointed out that if one makes adjustment with the modulation parameters so as to make the reacclerated spectrum consistent with the proton data, the same modulation parameters would predict too low a flux of helium nuclei. Therefore, we
2 10
near are
~‘
~ Nb1
~
~,
~ E~ 1~o -
conclude at this stage that reacceleration of cosmic rays in inter— stellar space would be inconsistent with the observed spectra of proton and helium nuclei. It may be seen from the parameters used by these authors for the escape time from the confinement volume,that low energy particles spent most of their time in the Galaxy. If one assumes that the energy dependent con-
~ c.
~
SO1J~MAXIMUM
~ io—l
100
101
ENERGY PER NUCLEON
io2
C GeV
Fic~. 2. Helium energy spectrum near the Earth based on re-acceleration models are shown by solid and dashed curves. Curve marked with
circles
represent
the observed spectrum.
(12)96
S. A. Stephens and R. L. Golden
finement Is apllicable only above certain rigidity R 0, it is possible to suppress the enhanced flux below about a few GeV. However, this procedure may not affect the spectral shape above about 10 GeV. Our calculations show that the resultant spectral shape of the proton spectrum in the energy region between 10 and 100 GeV is steeper by an index of about 0.1 compared to that of helium nuclei. One may be able to test this small difference with the future experiments. . Acknowledgement. This work is supported by NASA grant NGWA 110. REFERENCES -
~1.,237, 793 (1980).
1.
R.D. Blanford and J.P. Ostriker, Astrophys.
2.
H. Moraal and W.I. Axford, Astron. Astrophys., 125, 204 (1983).
3.
R. Cowsik, Astrophys.
4.
J.L. Osborne and V.S. Ptuskin, Proc. (1987).
5.
M. Simon, 14. Heinrich and K.D. Mathais, Astrophys.
6.
M. Simon, U. Heinbach and Ch. Koch, Astrophys. .1., 320, 699 (1987).
7.
A. Wandel, D. Eichler, J.R. Letaw, Astrophys. L. 316, 676 (1987).
8.
M. Giler, B. Szabelski, J. Wodwczyk and A.W. Wolfendale, Proc. 2Q..tXI Cosmic ~y Conf., 2, 211 (1987).
.In~L
9.
Ph. Ferrando and A. Soutoul, Proc. ~.Q~jjInt. Cosmic B~iConf., (1987)
2,
231
and
A.
L. 241, 1195 (1980). 2Q.~cJI
R.
~
Cosmic
~
Conf., 2, 218
L~ 300, 32 (1986).
Silberberg
and
C.S.
10. 3.R. Letaw, R. Sllberberg, C.H. Tsao, D. Elchler, M.M. Shapiro Wandel, Proc. 2.Q..th jj~ Cosmic ~y Conf., 2, 222 (1987).
Tsao,
11.. J.S. Perko, Astron. Astrophys., 184, 119 (1987). 12. S.A. Stephens and R.L. Golden, this issue. 13. W.R. Webber, R.L. Golden and S.A. Stephens, Proc. Conf., 1, 325 (1987).
2.Q.~Jj ~
Cosmic
R~