6/26/2006 David Gerstle 1

Muon Cosmic Ray Flux Study

David Gerstle

LArTPC – Yale University Undergraduate

 

6/26/2006 David Gerstle 2

Contents• CAPRICE 94 & 97 data and calculations

– Includes comparison of data to other trials and– an attempt at correcting for atmospheric depth

differences• Muon track length in LAr and calculations

related to CAPRICE 94 & 97 data.• The above applied to the Soudan, MN

atmospheric depth and (sort of) geomagnetic cutoff

• Conclusions• Other plots

6/26/2006 David Gerstle 3

What are the data?• Data are taken from Kremer et al., "Measurements of Ground-Level Muons at

Two Geomagnetic Locations," PRL 1999, Vol 83, No. 21, pp. 4241• CAPRICE is a spectrometer including:

– A ring imaging Cherenkov (RICH) detector– A time-of-flight system– A superconducting magnet spectrometer – A 7 rad length silicon-tungsten imaging calorimeter.

• Data were taken twice [“Data” tab of Cosmic Rays.xls; see DocDb]:– CAPRICE94: In 1994 at Lynn Lake, Manitoba, Canada (56.5° N, 101.0° W), 360m

altitude, a vertical geomagnetic cutoff of 0.5GeV, and atmospheric depth of 1000 g/cm^2.

– CAPRICE97: In 1997 at Fort Summer, New Mexico (34.4 ° N, 104.1° W), 1270m altitude, a cutoff of 4.2GeV, and depth of 886 g/cm^2.

• The data are given as dN/dp (including uncertainty) as a function of momentum. Data are provided in bins of varying width. A mean momentum for each bin is provided.– mu_+ and mu_- data are provided for each study.– The 97 mu_+ 1.6-2.1 GeV/c point was added by linear interpolation.

6/26/2006 David Gerstle 4

What did I do with them?• I copied them into Excel and then reorganized

them so they were graphable [“Multiplied Out”], etc. (no longer w/ scaling factors).

• Using a rectangular method over each bin I calculated the integral of (dN/dp)*dp [“Integral”], giving us N.– This is the ‘best’ method as data were collected and

provided in bins of varying width.• I plotted dN/dp as a function of p and N (from the

integral) as a function of p on the same plot for each data set AND summed the mu_+/mu_- data for each dataset into one plot.

6/26/2006 David Gerstle 5

CAPRICE94 mu_+

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

5

10

15

20

25

30

35

40

45

50

N (

as

inte

gra

l fro

m ~

0 t

o p

) (m

^2

sr

s)^

-1

[“95 mu_+”]

Integral curve is value from 0 GeV/c to plotted p

Blue is differential flux (left logarithmic y-axis)

Pink is integral flux from 0 GeV/c to p (right linear y-axis)

6/26/2006 David Gerstle 6

CAPRICE94 mu_-

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

5

10

15

20

25

30

35

40

45

50

N (

as

inte

gra

l fro

m ~

0 t

o p

) (m

^2

sr

s)^

-1

More mu_+ than mu_-

[“94 mu_-”]

Blue is differential flux

Pink is integral flux from 0 GeV/c to p

6/26/2006 David Gerstle 7

CAPRICE97 mu_+

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

6

12

18

24

30

36

42

48

54

60

N (

as

inte

gra

l fro

m ~

0 t

o p

) (m

^2

sr

s)^

-1

[“97 mu_+”]

Blue is differential flux

Pink is integral flux from 0 GeV/c to p

These are 97 data…

6/26/2006 David Gerstle 8

CAPRICE97 mu_-

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

5

10

15

20

25

30

35

40

45

50

NN

(a

s in

teg

ral f

rom

~0

to

p)(

m^

2 s

r s

)^-1

[“97 mu_-”]

Blue is differential flux

Pink is integral flux from 0 GeV/c to p

Again, more mu_+ than mu_-

6/26/2006 David Gerstle 9

CAPRICE 94&97 all data

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV/c

m^2

sr

s)^-

1

0

22

44

66

88

110

N (

as in

teg

ral f

rom

~0

to p

) (m

^2 s

r s)

^-1

94 data

97 data

94 integral

97 integral

Summing mu_+ and mu_- data, 97 appear higher than 94

[“94&97”]

6/26/2006 David Gerstle 10

Ratios of Data: 97/94

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

0.1 1 10 100

p; GeV/c

Ra

tio

of

Da

ta

97 data really are higher than 94 outside of uncertainty…why?

[“Ratio Graph”]

6/26/2006 David Gerstle 11

Differential Flux Adjusted to Account for Atmospheric Depth

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; (GeV/c)

dN

/dp

; (G

eV

/c m

^2

s s

r)^

-1

94 Adjusted

97

94 Original

Atmospheric depth effect described for 0.3<p<8 GeV/c by Attenuation Length Λ(g/cm^2)= (269±17)+(180±17)*p [M. Boezio, thesis, KTH]. Data are adjusted by e-(x97-x94)/Λ

The remaining low momentum difference can be accounted for by the geomagnetic cutoff on the incident protons (higher cutoff lowers flux)

[“Depth Graph”]

6/26/2006 David Gerstle 12

Data do match other studies

CAPRICE94&97 data compaired with PDG data

0.001

0.01

0.1

1 10 100 1000

p (GeV/c)

p^

2.7

dN

/dp

(c

m^

2 s

sr)

^-1

(Ge

V/c

)^1

.7

Blue is 94 differential flux

Pink is 97 differential flux

Yellow is from PDG cosmics plot

6/26/2006 David Gerstle 13

What are the track lengths in LAr?

• From “Atomic Nuclear Properties: Table 289: Muons in liquid argon” [“Track Length”]– Given as a density dependent -dE/dx (MeV cm^2/g) at various

energy/momentum values; converted into pure dE/dx.– Integrated (-dx/dE)*dE by a trapezoidal approximation to find the

distance traveled between energy points given by the chart.– Summed these to find the integral (-dx/dE)*dE from 0 GeV to a

given GeV (which is easily converted to p rather than E).• Put these data into the momentum bins of the CAPRICE

data.– Interpolated data for CAPRICE momentum bins without Table

289 data by a least-squares quadratic approximation (w/ four points) in relation to Table 289 data already in CAPRICE bins.

– Interpolated one point linearly from the Table 289 data.– The curve looks smooth (so I think I interpolated correctly):

6/26/2006 David Gerstle 14

Mean Track Length Over Bin (m)

0

10

20

30

40

50

60

0.1 1 10 100

p; (GeV/c)

Len

gth

; (m

)

•End plateau accounts for finite detector size

•Some data are interpolated, and it is good that you cannot tell which

[“Length”]

6/26/2006 David Gerstle 15

•Length at each momentum is multiplied by the differential flux; pink is 97, blue is 94.

•Yellow and Teal are the areas under the blue and pink curves (rectangular over each bin) from zero to p.

[“Length and Flux”]

Product of Differential Flux and Length AND Total Length through an Area, both as a function of Momentum

0

20

40

60

80

100

120

140

160

180

0.1 1 10 100

p; (GeV/c)

(dN

*x(p

))/d

p;

(GeV

/c m

s s

r)^-

1

0

200

400

600

800

1000

1200

1400

1600

N*x

(p)

fro

m z

ero

to

p;

(m s

sr)

^-1

94 (dN*x(p))/dp

97 (dN*x(p))/dp

94 N*x(p) from zero to p

97 N*x(p) from zero to p

Bumps must be results of flux data

6/26/2006 David Gerstle 16

What is the Bottom Line?

• By averaging the values from the data sets [“BOTTOM LINE”] :– The choice of 4 sig figs is arbitrary– Total N = 93.77 (m^2 s sr)^-1– Total N*x(p) = 1267.5 m*(m s sr)^-1– Average x = 13.54 m*(m^2 s sr)^-1

• Photons next…

6/26/2006 David Gerstle 17

Atmospheric Depth and Geomagnetic Cutoff Corrections

• Atmospheric depth of Soudan, MN was interpolated by a linear fit of atmospheric depth as a function of altitude between the CAPRICE 94&97 data.

• Calculations are in “Soudan.xls”• Atmospheric Depth was corrected for as outlined in M.

Boezio, thesis, KTH– Attenuation Length Λ(g/cm^2)= (269±17)+(180±17)*p– Data are adjusted by e-(xSoudan-x94)/Λ and e-(xSoudan-x97)/Λ

• Geomagnetic cutoff corrected for by a simple average between the two sets of corrected CAPRICE data.– The latitude of Soudan (47.8°) is nearly halfway between the

CAPRICE latitudes (56.5° and 34.3°).– There is a direct correlation between latitude and

geomagnetic cutoff, but I do not know exactly what it is. Read: this is a very crude correction.

6/26/2006 David Gerstle 18

CAPRICE differential flux adjusted to Soudan Atmospheric Depth

1

10

100

0.1 1 10

p; (GeV/c)

dN

/dp

; (G

eV

/c m

^2

s s

r)^

-1

Blue is 94 to Soudan

Pink is 97 to Soudan

Yellow is 94

Black is 97

Looks like a good correction; there is some discrepancy below .5 GeV/c, but this is likely because of the different geomagnetic cutoffs of 94&97

6/26/2006 David Gerstle 19

Soudan Differential and Integral Flux as a function of Momentum

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; (GeV/c)

dN

/dp

; (G

eV

/c m

^2

s s

r)^

-1

0

10

20

30

40

50

60

70

80

90

100

N f

rom

0 t

o p

; (m

^2

s s

r)^

-1

Looks like a nice curve; probably a good correction.

Note that integral flux is from 0 to p and is on the right-hand linear y-axis

Blue is differential flux

Pink is integral flux

6/26/2006 David Gerstle 20

Mean Track Length Over Bin (m)

0

10

20

30

40

50

60

0.1 1 10 100

p; (GeV/c)

Len

gth

; (m

)

Please recall this track length (referred to as x(p)) calculation as I apply it to Soudan…

6/26/2006 David Gerstle 21

(dN*x(p))/dp and N*x(p) as a function of p

0

20

40

60

80

100

120

140

0.1 1 10 100 1000

p; (GeV/c)

(dN

*x(p

))/d

p;

(Ge

V/c

m s

sr)

^-1

0

200

400

600

800

1000

1200

1400

N*x

(p);

(m

s s

r)^

-1

Blue is differential

Pink is integral

6/26/2006 David Gerstle 22

Average x(p) from 0 to p

0

2

4

6

8

10

12

14

16

0.1 1 10 100 1000

p; GeV/c

x;

m

Average is calculated by dividing the integral flux*length (N*x(p)) by the integral flux (N).

6/26/2006 David Gerstle 23

Soudan (sort of) BOTTOM LINE

• Integral (dN/dp)*dp = N from 0.2 GeV/c to 120 GeV/c = 86.92 (m^2 s sr)^-1

• Integral ((dN*x(p))/dp)*dp = N*x(p) from 0.2 GeV/c to 120 GeV/c = 1218.62 (m s sr)^-1

• N*x(p)/N = average x(p) = 14.02 m.• It would be good to find out the exact

atmospheric depth and geomagnetic cutoff.

6/26/2006 David Gerstle 24

6/26/2006 David Gerstle 25

Other plots:

6/26/2006 David Gerstle 26

CAPRICE94 Combined

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

N (

as

inte

gra

l fro

m ~

0 t

o p

)(m

^2

sr

s)^

-1

[“94”]

6/26/2006 David Gerstle 27

CAPRICE97 Combined

0.001

0.01

0.1

1

10

100

0.1 1 10 100 1000

p; GeV/c

dN

/dp

; (G

eV

/c m

^2

sr

s)^

-1

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

N (

as

inte

gra

l fro

m ~

0 t

o p

) (m

^2

sr

s)^

-1

[“97”]