宇宙航空研究開発機構研究開発報告 JAXA Research and Development Report

宇宙航空研究開発機構研究開発報告

JAXA Research and Development Report

JAXA極超音速風洞における6分力計測試験に係る 誤差評価について

Error Evaluation in JAXA Hypersonic Wind Tunnel Force Measurement Tests

藤井 啓介，高間 良樹

FUJII Keisuke and TAKAMA Yoshiki

2020年7月

宇宙航空研究開発機構

Japan Aerospace Exploration Agency

2 ޡࠩධՁʹؔ͢Δ੔ཧɾఆࣜԽ 8

2.1 ܭଌޡࠩͷۭྗ܎਺ਪఆޡࠩ΁ͷ఻೻. . . . 8

2.2 ؾྲྀͷඇҰ༷ੑ͕ٴ΅͢6෼ྗܭଌ΁ͷӨڹ . . . . 12

3 ޡࠩධՁͷ෩ಎࢼݧ΁ͷద༻ 16 3.1 JAXAۃ௒Ի଎෩ಎͱ6෼ྗܭଌܥͷ֓ཁ . . . . 16

3.2 ཁૉܭଌޡࠩ఻೻ . . . . 16

3.2.1 ఈ໘ѹpbܭଌޡࠩͷਪఆ . . . . 17

3.2.2 ϐτʔѹൺ^p_p⁰²_o ޡࠩͷਪఆ . . . . 19

3.2.3 HWT2ʹ͓͚Δඪ४໛ܕࢼݧʹΑΔݕূ . . . . 23

3.3 ඇҰ༷ੑޮՌਪఆͷ࣮ݧ݁ՌΛ༻͍ͨݕূ . . . . 26

3.3.1 ෩ಎࢼݧ֓ཁ(HWT14-51). . . . 26

3.3.2 ඇҰ༷ੑޮՌਪఆͷݕূ . . . . 28

3.3.3 HWT2ʹ͓͚ΔඇҰ༷ੑޮՌͷਪఆ . . . . 30

3.4 ࣠ରশܗঢ়ඪ४໛ܕΛ༻͍ͨ֯౓ޡࠩͷఱടग़ྗ΁ͷӨڹධՁ . . . . 30

3.4.1 ࠲ඪܥͷఆٛ . . . . 34

3.4.2 HWT2ඪ४໛ܕࢼݧ݁Ռ࠶੔ཧ . . . . 34

3.4.3 HWT1ඪ४໛ܕࢼݧ݁Ռ࠶੔ཧ . . . . 39

4 ݁࿦ 41 A HWT2ʹ͓͚Δ୅දҰ༷ྲྀMach਺ɺඇҰ༷ੑͱɺෆ͔֬͞ 45 B ఱടग़ྗ͔Βͷࢼݧ໛ܕॏྔॏ৺ٴͼఱടऔ෇֯ͷਪఆ 46 B.1 ҹՃՙॏͱఱടग़ྗ . . . . 48

B.2 ఱടॲཧʹ͓͚Δม਺౳ͷఆٛ . . . . 48

B.3 ॏྔɾॏ৺ɾΦϑηοτྔٴͼॳظ࢟੎֯ޡࠩਪఆ . . . . 51

B.4 ௨෩࣌ॲཧ . . . . 54

5

C ఱട֯౓ޡࠩʹΑΔޡࠩ఻೻ 54

6 6 10

14 14 14 15 17 21 24 24 26 28 28 32 32 37

39 43

44 46 46 49 52

52

This document is provided by JAXA

౻Ҫܒհɺߴؒྑथ

Error Evaluation in JAXA Hypersonic Wind Tunnel Force Measurement Tests

Fujii Keisuke and Takama Yoshiki

Abstract

The free-stream Mach number of JAXA 1.27m/0.5m hypersonic wind tunnel had been reported to actually fluctuate between blows with a small extent which could still be one of the major error sources in aerody- namic coefficients reduced from measurements. The cause of the fluctuation still remains unclear, however, an attempt to remedy such the unexpected varying Mach number effect has been made by accounting repre- sentative dynamic pressure variation between blows with leveraging pitot pressure monitoring measurement which is originally for wind tunnel validity evaluation purposes. The effectiveness of the remedy has then been evaluated based on data acquired through HB2 standard model tests ever conducted in the facility, in accordance with the error propagation relation developed for the proposed procedure to determine the free-stream Mach number.

Non-uniformity of the free-stream, however, cannot be accounted as an error source in the above uncertainty analysis, though previous M5 tests in the 0.5m hypersonic wind tunnel showed it actually has considerable effects, especially on the pitching moment coefficient. The mathematical formulation for estimating the error induced by the non-uniformity was then derived in terms of the three-dimensional spatial power spectrum of the pitot pressure ratio distribution and of the test model shape information, separately. Standard model test results where the test model was placed in different locations in the free-stream core region, agreed well with the estimated error from the non-uniformity, especially in the longitudinal aerodynamic characteristics, exhibiting the validity of the estimation formulation.

One of major error sources in the lateral-directional characteristics in the facility, however, turned out to be alignment errors of the balance coordinate and of the model coordinate, after examining again aerodynamic characteristics obtained in the axisymmetric standard model and reorganizing them by correcting the model coordinate directions based on the balance alignment angles estimated from static tare data.

keywords : Error in force measurements, Mach number fluctuation, Non-uniformity, Alignment error

֓ཁ

JAXA 1.27m/0.5mۃ௒Ի଎෩ಎࢼݧ6෼ྗܭଌʹ͓͚ΔޡࠩཁҼͱͯ͠Ұ༷ྲྀMach਺͕มಈ͢Δݱ৅ͷ֬

ೝ͞Εͨ͜ͱΛड͚ɺؾྲྀ݈શੑ֬ೝͷͨΊͷϐτʔѹͷϞχλʔܭଌ݁ՌΛར༻͠ɺ͜Ε·ͰෆมͱԾఆ͠

ͯॲཧ͞Ε͍ͯͨҰ༷ྲྀMach਺Λɺ௨෩ຖʹਪఆɾमਖ਼͢Δ͜ͱͰಈѹਪఆٴͼۭྗ܎਺ܭଌਫ਼౓ͷ޲্Λ

藤井 啓介 ，高間 良樹

Error Evaluation in JAXA Hypersonic Wind Tunnel Force Measurement Tests

FUJII Keisuke

^*1

, TAKAMA Yoshiki

^*1

概要

ࢼΈͨɻͦͷࡍʹैདྷσʔλॲཧٴͼ͜ͷಈѹमਖ਼ॲཧʹଈͨ͠ޡࠩ఻೻Λ੔ཧͨ͠͏͑Ͱɺܭଌਫ਼౓΁ͷޮ

ՌͷݕূΛ͜Ε·ͰܧଓతʹߦΘΕ͖͍ͯͯͨHB2ඪ४໛ܕࢼݧͷ݁ՌΛ΋ͱʹߦͬͨɻಉ࣌ʹɺJAXAۃ௒

Ի଎෩ಎͰͷඪ४తܭଌʹ͓͚ΔؾྲྀಛੑɾਪఆޡࠩɺܭଌޡࠩΛ੔ཧ͠Ұൠͷ6෼ྗܭଌࢼݧʹ͓͍ͯظ଴

͞ΕΔޡࠩਪఆํ๏Λ੔ཧ͠ɺͦͷଥ౰ੑ֬ೝΛߦͬͨɻߋʹؾྲྀͷඇҰ༷ੑʹΑΓൃੜ͢ΔۭؾྗɾϞʔϝ ϯτܭଌʹ͓͚Δෆ͔֬͞Λɺؾྲྀಈѹ෼෍ͷۭؒύϫʔεϖΫτϧͱɺ໛ܕܗঢ়৘ใͱ͔Β༧ଌ͢Δख๏Λ

։ൃ͠ɺͦͷ༧ଌଥ౰ੑΛඪ४໛ܕࢼݧ݁ՌΛجʹධՁͨ͠ɻͦͷ݁Ռɺಛʹॎ3෼ྗʹ͓͚ΔඇҰ༷ੑޮՌ Λ༧ଌ͢Δ͜ͱ͕Ͱ͖Δ͜ͱΛ֬ೝͨ͠ɻԣɾํ޲ಛੑʹؔ͢Δۃ௒Ի଎෩ಎࢼݧʹ͓͚ΔओཁͳޡࠩཁҼ͸

ҰํͰɺҰൠʹۭؾྗɾϞʔϝϯτͷେ͖͕͞খ͍ͨ͞ΊɺಈѹมಈͰ΋ɺඇҰ༷ੑͷޮՌͰ΋ͳ͘ɺఱട΍

໛ܕ࠲ඪͷऔ෇ޡࠩʹىҼ͢Δɺଞ෼ྗͷ࿙ΕͰ͋Δ৔߹ͷଟ͍͜ͱ͕ߟ͑ΒΕͨͨΊɺ෩ಎ໛ܕΛऔΓ෇͚

ͨঢ়ଶͰɺఱടʹՃΘΔॏྗΛར༻ͨ͠ఱടऔ෇֯౓ਪఆํ๏Λ৽ͨʹ։ൃ͠ɺաڈʹ࣮ࢪ͞Εͨ࣠ରশܗঢ় ͷඪ४໛ܕࢼݧʹద༻ͨ͠ɻͦͷ݁ՌɺఱടऔΓ෇͚֯౓ਪఆ݁ՌΛجʹ໛ܕͷؾྲྀʹର͢Δ࢟੎Λิਖ਼͢Δ

͜ͱͰɺԣɾํ޲ಛੑʹؔ͢ΔओཁޡࠩཁҼͱͯ͠ɺఱടɾ໛ܕऔΓ෇͚֯౓ਫ਼౓΋͋ΓಘΔ͜ͱΛ֬ೝͨ͠ɻ

ه߸

A ४Ұ࣍ݩྲྀʹ͓͚Δྲྀ࿏அ໘ੵ , m²

A2, A3, A4 ࠲ඪม׵3×3ߦྻ(ࣜ(27)ʹΑΔʣ , ND

A2p, A3p, A4p ࠲ඪม׵3×3ߦྻ(ࣜ(26)ʹΑΔʣ , ND AF ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , N⁻¹ Aijk ఱടֱਖ਼܎਺

AM ໛ܕ࠲ඪɾଌఆࣨ࠲ඪͷ࠲ඪม׵3×3ߦྻʢࣜ(29)ʣ

Ap02 ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , ND Apb ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , Pa⁻¹ Apo ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , Pa⁻¹ a0, a1, a2 ࣜ(24)ͰݱΕΔ܎਺ , ND ap ؾྲྀίΞ୅දϐτʔѹ/ϞχλʔܭଌҐஔϐτʔѹ(≡p^p02s⁰²) , ND a^∗_p ಛఆྖҬ಺ฏۉϐτʔѹൺͱಛఆҐஔϐτʔѹൺͱͷൺ཰ , ND Bijk ఱടֱਖ਼܎਺ʢࣜ(25)ʹΑΔ)

CAF લ໘࣠ྗ܎਺ , ND

CN B ϤʔΠϯάϞʔϝϯτ܎਺ɹ , ND

CY ԣྗ܎਺ , ND

Cp ѹྗ܎਺ , ND

co ໛ܕ࠲ඪʹ͓͚Δॏ৺ҐஔྻϕΫτϧ≡(cx, cy, cz)^t , m

cx, cy, cz ॏ৺Ґஔx, y, z࠲ඪ , m

C_{N B}^′ ༗ޮܴ֯ํ޲ʹରԠ͢ΔϤʔΠϯάϞʔϝϯτ܎਺(ࣜ(22)ʣ , ND C_Y^′ ༗ޮܴ֯ํ޲ʹରԠ͢Δԣྗ܎਺(ࣜ(21)ʣ , ND

D ໛ܕ୅ද௚ܘ , m

Db 3×6ఱടᎡΈ܎਺ߦྻɹʢࣜ(30)ʹΑΔʣ

Df ఱട࢟੎֯౳ਪఆ࣌F ͷ6×6ॏΈର֯ߦྻ , ND F ྗۭؒϕΫτϧͷۭؒϑʔϦΤม׵͞Εͨ΋ͷ , Nm³

ࢼΈͨɻͦͷࡍʹैདྷσʔλॲཧٴͼ͜ͷಈѹमਖ਼ॲཧʹଈͨ͠ޡࠩ఻೻Λ੔ཧͨ͠͏͑Ͱɺܭଌਫ਼౓΁ͷޮ

ՌͷݕূΛ͜Ε·ͰܧଓతʹߦΘΕ͖͍ͯͯͨHB2ඪ४໛ܕࢼݧͷ݁ՌΛ΋ͱʹߦͬͨɻಉ࣌ʹɺJAXAۃ௒

Ի଎෩ಎͰͷඪ४తܭଌʹ͓͚ΔؾྲྀಛੑɾਪఆޡࠩɺܭଌޡࠩΛ੔ཧ͠Ұൠͷ6෼ྗܭଌࢼݧʹ͓͍ͯظ଴

͞ΕΔޡࠩਪఆํ๏Λ੔ཧ͠ɺͦͷଥ౰ੑ֬ೝΛߦͬͨɻߋʹؾྲྀͷඇҰ༷ੑʹΑΓൃੜ͢ΔۭؾྗɾϞʔϝ ϯτܭଌʹ͓͚Δෆ͔֬͞Λɺؾྲྀಈѹ෼෍ͷۭؒύϫʔεϖΫτϧͱɺ໛ܕܗঢ়৘ใͱ͔Β༧ଌ͢Δख๏Λ

։ൃ͠ɺͦͷ༧ଌଥ౰ੑΛඪ४໛ܕࢼݧ݁ՌΛجʹධՁͨ͠ɻͦͷ݁Ռɺಛʹॎ3෼ྗʹ͓͚ΔඇҰ༷ੑޮՌ Λ༧ଌ͢Δ͜ͱ͕Ͱ͖Δ͜ͱΛ֬ೝͨ͠ɻԣɾํ޲ಛੑʹؔ͢Δۃ௒Ի଎෩ಎࢼݧʹ͓͚ΔओཁͳޡࠩཁҼ͸

ҰํͰɺҰൠʹۭؾྗɾϞʔϝϯτͷେ͖͕͞খ͍ͨ͞ΊɺಈѹมಈͰ΋ɺඇҰ༷ੑͷޮՌͰ΋ͳ͘ɺఱട΍

໛ܕ࠲ඪͷऔ෇ޡࠩʹىҼ͢Δɺଞ෼ྗͷ࿙ΕͰ͋Δ৔߹ͷଟ͍͜ͱ͕ߟ͑ΒΕͨͨΊɺ෩ಎ໛ܕΛऔΓ෇͚

ͨঢ়ଶͰɺఱടʹՃΘΔॏྗΛར༻ͨ͠ఱടऔ෇֯౓ਪఆํ๏Λ৽ͨʹ։ൃ͠ɺաڈʹ࣮ࢪ͞Εͨ࣠ରশܗঢ় ͷඪ४໛ܕࢼݧʹద༻ͨ͠ɻͦͷ݁ՌɺఱടऔΓ෇͚֯౓ਪఆ݁ՌΛجʹ໛ܕͷؾྲྀʹର͢Δ࢟੎Λิਖ਼͢Δ

͜ͱͰɺԣɾํ޲ಛੑʹؔ͢ΔओཁޡࠩཁҼͱͯ͠ɺఱടɾ໛ܕऔΓ෇͚֯౓ਫ਼౓΋͋ΓಘΔ͜ͱΛ֬ೝͨ͠ɻ

ه߸

A ४Ұ࣍ݩྲྀʹ͓͚Δྲྀ࿏அ໘ੵ , m²

A2, A3, A4 ࠲ඪม׵3×3ߦྻ(ࣜ(27)ʹΑΔʣ , ND

A2p, A3p, A4p ࠲ඪม׵3×3ߦྻ(ࣜ(26)ʹΑΔʣ , ND AF ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , N⁻¹ Aijk ఱടֱਖ਼܎਺

AM ໛ܕ࠲ඪɾଌఆࣨ࠲ඪͷ࠲ඪม׵3×3ߦྻʢࣜ(29)ʣ

Ap02 ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , ND Apb ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , Pa⁻¹ Apo ײ౓܎਺ʢࣜ(2)ʹΑΔʣ , Pa⁻¹ a0, a1, a2 ࣜ(24)ͰݱΕΔ܎਺ , ND ap ؾྲྀίΞ୅දϐτʔѹ/ϞχλʔܭଌҐஔϐτʔѹ(≡p^p02s⁰²) , ND a^∗_p ಛఆྖҬ಺ฏۉϐτʔѹൺͱಛఆҐஔϐτʔѹൺͱͷൺ཰ , ND Bijk ఱടֱਖ਼܎਺ʢࣜ(25)ʹΑΔ)

CAF લ໘࣠ྗ܎਺ , ND

CN B ϤʔΠϯάϞʔϝϯτ܎਺ɹ , ND

CY ԣྗ܎਺ , ND

Cp ѹྗ܎਺ , ND

co ໛ܕ࠲ඪʹ͓͚Δॏ৺ҐஔྻϕΫτϧ≡(cx, cy, cz)^t , m

cx, cy, cz ॏ৺Ґஔx, y, z࠲ඪ , m

C_{N B}^′ ༗ޮܴ֯ํ޲ʹରԠ͢ΔϤʔΠϯάϞʔϝϯτ܎਺(ࣜ(22)ʣ , ND C_Y^′ ༗ޮܴ֯ํ޲ʹରԠ͢Δԣྗ܎਺(ࣜ(21)ʣ , ND

D ໛ܕ୅ද௚ܘ , m

Db 3×6ఱടᎡΈ܎਺ߦྻɹʢࣜ(30)ʹΑΔʣ

Df ఱട࢟੎֯౳ਪఆ࣌F ͷ6×6ॏΈର֯ߦྻ , ND F ྗۭؒϕΫτϧͷۭؒϑʔϦΤม׵͞Εͨ΋ͷ , Nm³

F ྗɾϞʔϝϯτ6ݩྻϕΫτϧ , N·ͨ͸Nm

FA ࣠ྗ , N

Fi Fͷiํ޲੒෼ , Nm³

f ྗۭؒϕΫτϧ , N

fC ໛ܕʹ܎Δۭؾྗͷ೾਺ಛੑʢࣜ(13)ʣ , m² i,j,k ࠲ඪܥ௚ަ୯ҐϕΫτϧ , ND

K ෆ͔֬͞ղੳแׅ౓܎਺ , ND

KB ෆ͔֬͞ղੳόΠΞεภࠩͷแׅ౓܎਺ʹ૬౰͢Δ܎਺ , ND

L ୅ද௕͞ , m

Ld ৙ཚͷ୅ද௕͞εέʔϧ , m

Lm ڙࢼ໛ܕͷ୅ද௕͞εέʔϧ , m

M Ϛοϋ਺ , ND

M ϞʔϝϯτۭؒϕΫτϧͷۭؒϑʔϦΤม׵͞Εͨ΋ͷ , Nm⁴ M2B ໛ܕ࠲ඪɾఱട࠲ඪؒͷ࠲ඪม׵3×3ߦྻ(ࣜ(28)ʣ , ND

Mi Mͷiํ޲੒෼ , Nm⁴

m ϞʔϝϯτۭؒϕΫτϧ , Nm

mC ໛ܕʹ܎ΔۭྗϞʔϝϯτͷ೾਺ಛੑʢࣜ(15)ʣ , m³

p ѹྗ , Pa

p02 (୅දత)ϐτʔѹɹ , Pa

p02s ܭଌҐஔʹ͓͚Δϐτʔѹɹ , Pa

pT.C. ଌఆࣨѹྗ , Pa

Qf ಈѹqͷۭؒϑʔϦΤม׵ , Nm

q ಈѹ , Pa

Runit ୯ҐReynolds਺ , m⁻¹

S ໘ੵ , m²

S ఱടग़ྗྻϕΫτϧ ≡(S1, . . . , S6)^t , V

Si ఱടग़ྗ , V

s ද໘ཁૉۭؒϕΫτϧ , m²

s 3ݩྻϕΫτϧɹ≡(sx, sy, sz)^t , ND

sx ≡sinθg , ND

sy ≡cosθgsinϕg , ND

sz ≡ −cosθgcosϕg , ND

t ࣌ؒ , s

U ෆ͔֬͞

V ମੵ , m³

W ॏྔ , N

Wc ॏྔॏ৺ʹؔ͢Δ4ݩྻϕΫτϧ≡(W, cx, cy, cz)^t , Nຢ͸m

x ҐஔۭؒϕΫτϧ , m

xo ໛ܕݪ఺ͷҐஔۭؒϕΫτϧ , m

x, y, z xͷ࠲ඪ੒෼ , m

α ܴ֯ , deg

α, β, γ ೾਺ , m⁻¹

αeff ༗ޮܴ֯ , deg

β ԣ׈Γ֯ , deg

γ ൺ೤ൺ

δ ޡࠩ

δ^∗ ਪఆޡࠩ

Θ (β, θ, ϕ)ྻϕΫτϧ , deg

Θg (θg, ϕg)2ݩྻϕΫτϧ , deg θ, ϕ ϐον֯ɺϩʔϧ֯ , deg

σ ඪ४ภࠩ

σf ྗͷඪ४ภۭࠩؒϕΫτϧ , N σm Ϟʔϝϯτͷඪ४ภۭࠩؒϕΫτϧ , Nm subscript

0 ਅ஋

b ఈ໘

o ఽΈঢ়ଶ

∞ Ұ༷ྲྀঢ়ଶ

CM Ϟʔϝϯτج४Ґஔ

superscript

′ มಈ੒෼

x, y, z xͷ࠲ඪ੒෼ , m

α ܴ֯ , deg

α, β, γ ೾਺ , m⁻¹

αeff ༗ޮܴ֯ , deg

β ԣ׈Γ֯ , deg

γ ൺ೤ൺ

δ ޡࠩ

δ^∗ ਪఆޡࠩ

Θ (β, θ, ϕ)ྻϕΫτϧ , deg

Θg (θg, ϕg)2ݩྻϕΫτϧ , deg θ, ϕ ϐον֯ɺϩʔϧ֯ , deg

σ ඪ४ภࠩ

σf ྗͷඪ४ภۭࠩؒϕΫτϧ , N σm Ϟʔϝϯτͷඪ४ภۭࠩؒϕΫτϧ , Nm subscript

0 ਅ஋

b ఈ໘

o ఽΈঢ়ଶ

∞ Ұ༷ྲྀঢ়ଶ

CM Ϟʔϝϯτج४Ґஔ

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′ มಈ੒෼

1 ং࿦

JAXAϕ0.5m/1.27mۃ௒Ի଎෩ಎ[1]ʹ͓͚Δݱঢ়ͷσʔλॲཧ͸ɺ෩ಎؾྲֱྀਖ਼ࢼݧ[2]ʹΑͬͯಘΒΕͨ

Ұ༷ྲྀMach਺͕ෆมͰ͋ΔͱԾఆ͠ɺܭଌ͞ΕΔ෩ಎఽΈѹྗpoɺఽΈԹ౓To͔Βɺฏߧ࣮ࡏؾମΛԾఆ

͢Δ͜ͱʹΑͬͯҰ༷ྲྀ੩ѹp_∞ٴͼಈѹqΛਪఆ͠ɺఱടग़ྗͰ͋ΔۭؾྗΛແ࣍ݩԽ͢Δख๏Λ࠾༻ͯ͠

͖͍ͯΔɻ͔͠͠ͳ͕ΒɺҰൠʹϊζϧ๲ுաఔʹ͓͍ͯٸ଎ʹݮগ͢Δີ౓ͱڞʹɺৼಈྭىΤωϧΪʔ͕

ࣄ্࣮ౚ݁ͯ͠͠·͏͍ΘΏΔnozzle freezing͕εϩʔτۙ๣Ͱൃੜ͢Δ͜ͱ͕༧ଌ͞Ε͓ͯΓ[3]ɺ࣮ࡍɺۙ

೥ͷ෩ࢼσʔλͷ࠶੔ཧʹ͓͍ͯฏߧ࣮ࡏؾମΛԾఆͨ͠ϊζϧ๲ுؔ܎ࣜΑΓ΋ϊζϧౚ݁Λ໛ٖ͢Δ׬શ ؾମؔ܎ࣜͷํ͕ɺೡΖۭྗಛੑଌఆ஋͹Β͖ͭΛ௿ݮͤ͞Δͱ͍͏ใࠂ͕ͳ͞Ε͍ͯΔ[4]ɻͦͷόϥ͖ͭධ Ձͷࡍʹ͜Ε·Ͱͷ෩ಎࢼݧʹ͓͍ͯҰ༷ྲྀ੒ཱੑ֬ೝͷ໨తͰಉ࣌ܭଌ͍ͯͨ͠ϐτʔѹϞχλʕܭଌ஋Λ

࠶੔ཧͨ͠ͱ͜Ζɺ௨෩ຖʹϐτʔѹൺp02/po͕ແࢹͰ͖ͳ͍ϨϕϧͰมಈ͍ͯ͠Δ͜ͱ͕͋Θͤͯ֬ೝ͞Ε

ͨɻ͜Ε͸ɺෆมͰ͋ΔͱԾఆ͞Ε͍ͯͨҰ༷ྲྀMach਺͕౎౓มಈ͍ͯ͠Δ͜ͱɺ·ͨͦͷมಈྔ͕࣮ݧ݁

Ռͷ࠶ݱੑݕূʹ͓͍ͯେ͖ͳӨڹΛ༩͑Δఔ౓Ͱ͋Δ͜ͱΛҙຯ͍ͯͨ͠ɻ͜Ε·Ͱͷ6෼ྗܭଌʹ͓͚Δ ޡࠩධՁͰ͸ɺؾྲֱྀਖ਼ࢼݧ࣌ʹಘΒΕͨMach਺ͷۭؒతɾ࣌ؒతมಈྔͱɺ֤ܭଌྔʹ͓͚ΔޡࠩΛ૊Έ߹

ΘͤͨͷΈͰ͋ΓɺࢼݧΩϟϯϖʔϯຖʹมಈ͢Δϐτʔѹൺp02/poʢ͋Δ͍͸ɺҰ༷ྲྀMach਺)ʹؔ͢Δ ߟྀ͕ͳ͔ͬͨͨΊɺঢ়گʹΑΓ૝ఆ͞ΕΔޡࠩൣғҎ্ͷ࠶ݱੑࠩҟͷੜ͡Δ͜ͱ͕༧ଌ͞Εɺ࣮ࡍಉҰ໛

ܕʹΑΔҟͳΔ࣌ظʹ࣮ࢪͨ͠ࢼݧΩϟϯϖʔϯʹ͓͍ͯ૝ఆޡࠩҎ্ͷ࠶ݱੑόϥ͖ͭͷ͋Δ͜ͱ͕෼͔ͬ

ͨɻͦ͜Ͱܭଌ੍໿͕େ͖͍΋ͷͷϐτʔѹϞχλʔܭଌΛ༻͍ࢼݧຖͰมಈ͢Δ෺ཧྔΛ‘ܭଌޡࠩ’ͱͯ͠

ѻ͏ͷͰ͸ͳ͘ਖ਼౰ʹධՁ͢Δ͜ͱͰɺΑΓਫ਼౓ͷߴ͍ࢼݧɾܭଌΛ໨ࢦ͢͜ͱͱͨ͠ɻͦͷΑ͏ͳσʔλॲ ཧΛ͢Δ৔߹ɺ֤ʑͷܭଌޡࠩɺ࠶ݱੑόϥ͖ͭͷ6෼ྗ’ܭଌ஋’ޡࠩ΁ͷ఻೻Λ໌֬Խ͠ɺͦͷํ๏ʹΑΔ ޡࠩ௿ݮޮՌΛධՁ͢Δඞཁ͕ੜ͡ɺJAXAϕ1.27mۃ௒Ի଎෩ಎʹ͓͚Δैલͱ৽نఏҊख๏ʹΑΔσʔλ ॲཧͦΕͧΕʹଈͨ͠ޡࠩ఻೻Λ੔ཧͨ͠ɻ͜͜Ͱ͸ɺ6෼ྗۭྗ܎਺த࠷΋ؔ࿈෺ཧྔͷଟ͍લ໘࣠ྗ܎਺

CAFʹॏ఺Λஔ͖ɺաڈʹ࣮ࢪ͞Εͨඪ४໛ܕࢼݧٴͼؾྲֱྀਖ਼ࢼݧ౳Λجʹɺ૝ఆ͞ΕΔ֤ܭଌɾਪఆཁૉ

ʹ͓͚Δޡࠩͷجૅσʔλ੔ཧɾଥ౰ੑධՁΛߦ͏͜ͱͱͨ͠ɻ͜ͷ੔ཧʹΑΓ୯ಠͷ̒෼ྗࢼݧʹ͓͚Δޡ

ࠩྔਪఆʹඞཁͳ෩ಎಛੑʹؔ͢Δෆ֬ఆੑɺ֤ܭଌʹ͓͚ΔޡࠩྔΛ໌֬Խ͢Δ͜ͱΛ໨తͱͨ͠ɻ

ͦΕΒ֤ܭଌɾਪఆޡࠩ఻೻ͷݕ౼Ͱ͸෩ಎҰ༷ྲྀίΞྖҬ಺Ͱͷϐτʔѹฏۉ஋͓Αͼɺͦͷฏۉ஋ͷมಈ

͢ΔྔΛߟྀͨ͠ޡࠩධՁΛߦ͍ͬͯΔ͕ɺ6෼ྗܭଌ΁ͷಈѹ෼෍ͷޮՌɺ͋Δ͍͸ؾྲྀͷඇҰ༷ੑͷޮՌ Λਖ਼౰ʹධՁ͍ͯ͠Δͱ͸ݴ͑ͳ͍ɻؾྲྀͷඇҰ༷ੑʹىҼ͢Δۭྗಛੑ΁ͷޮՌʹؔͯ͠͸ɺ໛ܕҐஔΛൺ

ֱత༰қʹมߋͰ͖Δϕ0.5mۃ௒Ի଎෩ಎ(HWT1)ʹ͓͚Δඪ४໛ܕࢼݧʹΑΓಛʹϐονϯάϞʔϝϯτ

΁ͷӨڹ͕͔ͳΒͣ͠΋ແࢹͰ͖ͳ͍ྔͱͳ͍ͬͯΔ͜ͱ͕൑໌͍ͯ͠Δ[5]ɻ͔͠͠ͳ͕ΒͦͷඇҰ༷ੑͷޮ

Ռ͸໛ܕܗঢ়ʹڧ͘ґଘ͢Δ͜ͱ΋૝ఆ͞ΕΔͨΊɺඪ४໛ܕࢼݧ΁ͷޮՌΛಛఆͨ͠ͷΈͰ͸ɺҰൠͷࢼݧ ʹ͓͚ΔޡࠩධՁΛ͢Δ͜ͱ͕Ͱ͖ͳ͍ɻඇҰ༷ੑޮՌͷޡࠩݶքΛ஌ΔͨΊͷ౒ྗͱͯ͜͠Ε·Ͱʹɺಈѹ

෼෍ʹ͓͚Δ௕͞εέʔϧΛߟྀͤͣಈѹͷۭؒతมಈྔ͕ͦͷ··ۭؾྗͷภΓޡࠩͱͳΔͱ͍͏ϞσϧΛ

༻͍Δ؆қख๏͸࣮ࢪ͞ΕΔ͜ͱ͕͕͋ͬͨɺͦͷΑ͏ͳํ๏Ͱ͸ɺྗܭଌ΁ͷӨڹʹؔ͠ෆ͔֬͞ͷ্ݶΛ

཈͑Δ͜ͱ͸Ͱ͖Δ΋ͷͷɺϞʔϝϯτܭଌʹ͓͚Δෆ͔֬͞ʹؔͯ͠͸ɺ߹ཧతͳ௕͞εέʔϧΛߟྀ͠ͳ

͍ͨΊ্ݶ஋͢ΒධՁ͢Δ͜ͱ͸Ͱ͖͍ͯͳ͍ɻྫ͑͹ಈѹ෼෍ඇҰ༷ੑͷ୅ද௕͞εέʔϧLdʹ͘Β΂໛

ܕ୅ද௕Lm͕े෼ʹେ͖͍৔߹(Ld≪Lm)͸ඇҰ༷ੑͷӨڹ͸ฏۉԽ͞ΕແࢹͰ͖Δͱ༧૝Ͱ͖ΔҰํͰɺ ٯͷঢ়گͷ৔߹(Ld≫Lm)ɺܭଌ͞ΕΔۭؾྗͷόϥ͖ͭ͸ؾྲྀඇҰ༷ੑͷಈѹόϥ͖ͭͱಉఔ౓ͱͳΔ΋ͷ ͱ༧૝͞Εɺ୅ද௕͞εέʔϧ͕ಉఔ౓Lm≈LdͱͳΓͦ͏Ͱ͋Δ৔߹ɺӨڹ͸ͦͷதؒͷఔ౓ͱͳΔͱߟ

͑ΒΕΔͨΊɺ໛ܕ୅ද௕͞ͱಈѹ෼෍୅ද௕͞εέʔϧͱͷؔ܎ͰͦͷӨڹ͸มԽ͢Δ͸ͣͷ΋ͷͰ͋Δɻ ߋʹϞʔϝϯτʹؔͯ͠͸ɺLd ≪Lmͷ৔߹Ͱ΋Lm≪Ldͷ͍ͣΕͷ৔߹Ͱ΋໛ܕҐஔͷӨڹ͸ۃݶͰ͸

ফ໓͢Δ͜ͱͱͳΓɺLm≈Ldͷ෇ۙʹ͓͍ͯͷΈ༗ݶ஋ΛͱΔͱ༧૝Ͱ͖Δ͕ɺͦͷ্ݶ஋Λ؆ศʹಘΔ൚

༻తͳख๏͸͜Ε·ͰʹఏҊ͞Ε͍ͯͳ͍ɻ͜ͷ໰୊͸ɺ෩ಎࢼݧΛ௨ͯ͡τϦϜܴ֯ͷ༧ଌ΍ඞཁͳ଩໘ͷ

１　序論

ઃܭΛ͢Δঢ়گʹ͓͍ͯॏཁͳཁૉͱͳΔՄೳੑ͕ߟ͑ΒΕΔͨΊɺؾྲྀͷඇҰ༷ੑͷٴ΅ۭ͢ؾྗɺಛʹۭ

ྗϞʔϝϯτ΁ͷӨڹΛࢼݧ໛ܕʹԠͯ͡ద੾ʹ༧ଌ͢Δ͜ͱͷඞཁੑΛ͍ࣔͯ͠Δͱ͍͑ΔɻۭؾྗɾϞʔ ϝϯτ΁ͷӨڹ౓Λܾఆ͢Δͱߟ͑ΒΕΔؾྲྀඇҰ༷ੑͷ୅ද௕Ld͸೾਺α, β, γͷٯ਺ͱͱΒ͑Δ͜ͱ͕Ͱ

͖ɺ෩ಎؾྲྀಈѹ෼෍ͷۭؒύϫʔεϖΫτϧ͕ӨڹධՁͷͨΊʹඞཁͳ৘ใͰ͋Δͱߟ͑ΒΕͨɻͦͷͨΊ ؾྲֱྀਖ਼ࢼݧͰಘΒΕ͍ͯΔಈѹ෼෍ͷۭؒύϫʔεϖΫτϧΛͱΓɺ෼෍ͷ௕͞εέʔϧͱ໛ܕ୅ද௕ͱͷ

ؔ܎Λߟྀ͢ΔӨڹධՁͷఆࣜԽΛߦ͏͜ͱͱͨ͠ɻ·ͨͦͷධՁ๏ͷݕূͱͯ͠ɺաڈʹHWT1M5෩ಎʹ

͓͍࣮ͯࢪ͞Εͨ໛ܕઃஔҐஔมߋޮՌࢼݧ݁Ռʹ͓͚Δۭྗಛੑมಈྔͱɺ༧ଌ݁ՌͱͷൺֱΛߦ͍ɺͦͷ ଥ౰ੑΛ֬ೝ͢Δ͜ͱͱͨ͠ɻ

ۃ௒Ի଎෩ಎ6෼ྗܭଌʹ͓͚ΔओཁͳޡࠩཁҼ͸Ҏ্ʹΑΓධՁͰ͖Δͱߟ͑ΒΕΔ͕ɺ௨ৗͷࢼݧܗଶͰ

͸Ұൠʹۭؾྗͷখ͍͞ԣɾํ޲ಛੑʹؔͯ͠͸ɺఱടɾ໛ܕऔ෇֯ޡࠩΛ௨ͨ͠ΑΓେ͖ͳۭؾྗͷൃੜ͠

͍ͯΔॎ3෼ྗͷӨڹ͕ೡΖओཁͳޡࠩཁҼͱͳΓಘΔ͜ͱ͕൑໌͖ͯͨ͠ɻࢼݧ໛ܕऔ෇࣌౳ʹൃੜ͢Δ໛

ܕ࠲ඪܥͱఱട࠲ඪܥʹ͓͚Δऔ෇֯ޡࠩʹىҼ͢ΔޡࠩධՁʹ͓͍ͯɺࢼݧ໛ܕऔ෇ঢ়ଶʹ͓͚Δ໛ܕ࢟੎

֯ܭଌʹՃ͑ɺͦͷঢ়ଶʹ͓͚Δఱട࢟੎֯ܭଌΛߦ͏͜ͱͰɺ໛ܕɾఱടؒऔ෇֯౓Λਪఆ͢Δඞཁ͕͋Δɻ

ͦͷΑ͏ͳഎܠ͔Βۃ௒Ի଎෩ಎͰ͸ɺ໛ܕʹಇ͘ॏྗΛෳ਺࢟੎֯έʔεͰܭଌ͢Δ͜ͱͰɺఱടॲཧʹඞ ཁͳ໛ܕॏྔɾॏ৺Ґஔɺఱടग़ྗΦϑηοτྔͷਪఆʹՃ͑ɺఱട࠲ඪܥͷ࢟੎֯ਪఆ΋͋Θͤͯߦ͏ํ๏

Λۙ೥औΓೖΕ͍ͯΔɻ͜͜Ͱ͸͜ͷఱടऔ෇֯ਪఆͷଥ౰ੑΛධՁ͢Δ͜ͱΛ໨తʹɺ͜Ε·ͰʹߦΘΕͯ

͖ͨ࣠ରশඪ४໛ܕࢼݧۭྗಛੑٴͼఱടֱਖ਼݁ՌΛ࠶ධՁ͢Δ͜ͱͱͨ͠ɻͦΕʹ൐͍ɺ෩ಎࢼݧ࣌ʹ૝ఆ

͢΂͖໛ܕɾఱടऔ෇֯౓ޡࠩͱͯ͠ݟࠐΉ΂͖ྔΛਪఆ͢Δ͜ͱͱͨ͠ɻ·ͨɺຊߘʹ͓͍ͯɺޡࠩ͸ݸʑ ͷܭଌ஋ͱਅ஋ͱͷࠩΛࢦ͠ɺෆ͔֬͞͸ܭଌ஋पΓʹਅ஋ͷଘࡏ͢ΔൣғΛࢦ͢΋ͷͱ͢Δɻ

2 ޡࠩධՁʹؔ͢Δ੔ཧɾఆࣜԽ

2.1 ܭଌޡࠩͷۭྗ܎਺ਪఆޡࠩ΁ͷ఻೻

Ұൠʹɺલ໘࣠ྗ܎਺CAF ͸ܭଌྔ͋Δ͍͸ܭଌྔ͔ΒٻΊΒΕΔঢ়ଶྔΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͞Εͯ

͍Δ:

CAF ≡FA−Sb(p_∞−pb) qS

͜͜Ͱɺ෩ಎࢼݧʹ͓͚ΔܭଌྔͰ͋Δ࣠ྗFAٴͼϕʔεѹpb͸͍ͣΕ΋ɺҎԼͷΑ͏ʹ‘ਅ஋’ͱܭଌޡࠩ

ͱ͔Βߏ੒͞ΕΔͱߟ͑ɺ·ͨਅ஋ɺܭଌޡࠩ͸ͦΕͧΕฏۉ஋ٴͼมಈ੒෼ʹ෼͚ͯߟ͑Δ͜ͱ͕Ͱ͖Δ:

FA=FA,0+δFA

=FA,0+ (FA,0)^′+δFA+ (δFA)^′ pb=pb,0+δpb

=pb,0+ (pb,0)^′+δpb+ (δpb)^′

͜͜Ͱ͸ܭଌޡࠩ͸ਅ஋ʹର͠े෼খ͘͞ɺ·ͨͦΕͧΕͷฏۉ஋ʹର͠มಈ੒෼͸े෼ʹখ͍͞΋ͷͱߟ͑ɺ ඍখྔʹؔͯ͠ઢܕԽͯ͠੔ཧ͢Δ͜ͱͱ͢ΔɻҰํɺಈѹqɺҰ༷ྲྀ੩ѹp_∞͸௚઀ܭଌ͞ΕΔྔͰ͸ͳ͘ɺ ܭଌྔͰ͋Δϐτʔѹp02ٴͼ෩ಎఽΈѹྗpoͱɺͦΕΒ͔Βఆٛ͞ΕΔҰ༷ྲྀMach਺M ͱΛ༻͍ͯҎԼ

2 　誤差評価に関する整理・定式化

ઃܭΛ͢Δঢ়گʹ͓͍ͯॏཁͳཁૉͱͳΔՄೳੑ͕ߟ͑ΒΕΔͨΊɺؾྲྀͷඇҰ༷ੑͷٴ΅ۭ͢ؾྗɺಛʹۭ

ྗϞʔϝϯτ΁ͷӨڹΛࢼݧ໛ܕʹԠͯ͡ద੾ʹ༧ଌ͢Δ͜ͱͷඞཁੑΛ͍ࣔͯ͠Δͱ͍͑ΔɻۭؾྗɾϞʔ ϝϯτ΁ͷӨڹ౓Λܾఆ͢Δͱߟ͑ΒΕΔؾྲྀඇҰ༷ੑͷ୅ද௕Ld͸೾਺α, β, γͷٯ਺ͱͱΒ͑Δ͜ͱ͕Ͱ

͖ɺ෩ಎؾྲྀಈѹ෼෍ͷۭؒύϫʔεϖΫτϧ͕ӨڹධՁͷͨΊʹඞཁͳ৘ใͰ͋Δͱߟ͑ΒΕͨɻͦͷͨΊ ؾྲֱྀਖ਼ࢼݧͰಘΒΕ͍ͯΔಈѹ෼෍ͷۭؒύϫʔεϖΫτϧΛͱΓɺ෼෍ͷ௕͞εέʔϧͱ໛ܕ୅ද௕ͱͷ

ؔ܎Λߟྀ͢ΔӨڹධՁͷఆࣜԽΛߦ͏͜ͱͱͨ͠ɻ·ͨͦͷධՁ๏ͷݕূͱͯ͠ɺաڈʹHWT1M5෩ಎʹ

͓͍࣮ͯࢪ͞Εͨ໛ܕઃஔҐஔมߋޮՌࢼݧ݁Ռʹ͓͚Δۭྗಛੑมಈྔͱɺ༧ଌ݁ՌͱͷൺֱΛߦ͍ɺͦͷ ଥ౰ੑΛ֬ೝ͢Δ͜ͱͱͨ͠ɻ

ۃ௒Ի଎෩ಎ6෼ྗܭଌʹ͓͚ΔओཁͳޡࠩཁҼ͸Ҏ্ʹΑΓධՁͰ͖Δͱߟ͑ΒΕΔ͕ɺ௨ৗͷࢼݧܗଶͰ

͸Ұൠʹۭؾྗͷখ͍͞ԣɾํ޲ಛੑʹؔͯ͠͸ɺఱടɾ໛ܕऔ෇֯ޡࠩΛ௨ͨ͠ΑΓେ͖ͳۭؾྗͷൃੜ͠

͍ͯΔॎ3෼ྗͷӨڹ͕ೡΖओཁͳޡࠩཁҼͱͳΓಘΔ͜ͱ͕൑໌͖ͯͨ͠ɻࢼݧ໛ܕऔ෇࣌౳ʹൃੜ͢Δ໛

ܕ࠲ඪܥͱఱട࠲ඪܥʹ͓͚Δऔ෇֯ޡࠩʹىҼ͢ΔޡࠩධՁʹ͓͍ͯɺࢼݧ໛ܕऔ෇ঢ়ଶʹ͓͚Δ໛ܕ࢟੎

֯ܭଌʹՃ͑ɺͦͷঢ়ଶʹ͓͚Δఱട࢟੎֯ܭଌΛߦ͏͜ͱͰɺ໛ܕɾఱടؒऔ෇֯౓Λਪఆ͢Δඞཁ͕͋Δɻ

ͦͷΑ͏ͳഎܠ͔Βۃ௒Ի଎෩ಎͰ͸ɺ໛ܕʹಇ͘ॏྗΛෳ਺࢟੎֯έʔεͰܭଌ͢Δ͜ͱͰɺఱടॲཧʹඞ ཁͳ໛ܕॏྔɾॏ৺Ґஔɺఱടग़ྗΦϑηοτྔͷਪఆʹՃ͑ɺఱട࠲ඪܥͷ࢟੎֯ਪఆ΋͋Θͤͯߦ͏ํ๏

Λۙ೥औΓೖΕ͍ͯΔɻ͜͜Ͱ͸͜ͷఱടऔ෇֯ਪఆͷଥ౰ੑΛධՁ͢Δ͜ͱΛ໨తʹɺ͜Ε·ͰʹߦΘΕͯ

͖ͨ࣠ରশඪ४໛ܕࢼݧۭྗಛੑٴͼఱടֱਖ਼݁ՌΛ࠶ධՁ͢Δ͜ͱͱͨ͠ɻͦΕʹ൐͍ɺ෩ಎࢼݧ࣌ʹ૝ఆ

͢΂͖໛ܕɾఱടऔ෇֯౓ޡࠩͱͯ͠ݟࠐΉ΂͖ྔΛਪఆ͢Δ͜ͱͱͨ͠ɻ·ͨɺຊߘʹ͓͍ͯɺޡࠩ͸ݸʑ ͷܭଌ஋ͱਅ஋ͱͷࠩΛࢦ͠ɺෆ͔֬͞͸ܭଌ஋पΓʹਅ஋ͷଘࡏ͢ΔൣғΛࢦ͢΋ͷͱ͢Δɻ

2 ޡࠩධՁʹؔ͢Δ੔ཧɾఆࣜԽ

2.1 ܭଌޡࠩͷۭྗ܎਺ਪఆޡࠩ΁ͷ఻೻

Ұൠʹɺલ໘࣠ྗ܎਺CAF ͸ܭଌྔ͋Δ͍͸ܭଌྔ͔ΒٻΊΒΕΔঢ়ଶྔΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͞Εͯ

͍Δ:

CAF ≡FA−Sb(p_∞−pb) qS

͜͜Ͱɺ෩ಎࢼݧʹ͓͚ΔܭଌྔͰ͋Δ࣠ྗFAٴͼϕʔεѹpb͸͍ͣΕ΋ɺҎԼͷΑ͏ʹ‘ਅ஋’ͱܭଌޡࠩ

ͱ͔Βߏ੒͞ΕΔͱߟ͑ɺ·ͨਅ஋ɺܭଌޡࠩ͸ͦΕͧΕฏۉ஋ٴͼมಈ੒෼ʹ෼͚ͯߟ͑Δ͜ͱ͕Ͱ͖Δ:

FA=FA,0+δFA

=FA,0+ (FA,0)^′+δFA+ (δFA)^′ pb=pb,0+δpb

=pb,0+ (pb,0)^′+δpb+ (δpb)^′

͜͜Ͱ͸ܭଌޡࠩ͸ਅ஋ʹର͠े෼খ͘͞ɺ·ͨͦΕͧΕͷฏۉ஋ʹର͠มಈ੒෼͸े෼ʹখ͍͞΋ͷͱߟ͑ɺ ඍখྔʹؔͯ͠ઢܕԽͯ͠੔ཧ͢Δ͜ͱͱ͢ΔɻҰํɺಈѹqɺҰ༷ྲྀ੩ѹp_∞͸௚઀ܭଌ͞ΕΔྔͰ͸ͳ͘ɺ ܭଌྔͰ͋Δϐτʔѹp02ٴͼ෩ಎఽΈѹྗpoͱɺͦΕΒ͔Βఆٛ͞ΕΔҰ༷ྲྀMach਺M ͱΛ༻͍ͯҎԼ

ͷؔ܎ࣜʹΑΓ‘ఆٛ’͞ΕΔ΋ͷͱߟ͑Δ:

q≡po

γM² 2

1 +γ−1 2 M²

−γ−^γ1

p_∞≡po

1 +γ−1 2 M²

−_γ−1^γ

p02

po

=

1 + 2γ

γ+ 1(M²−1)

−_γ−1¹ (γ+ 1)M² (γ−1)M²+ 2

γ−^γ1

͜͜Ͱ͸ಈѹqɺҰ༷ྲྀ੩ѹp_∞͸Ϛοϋ਺M ͱؾྲྀఽΈѹྗpoͷؔ਺ͱͯ͠੔ཧ͓ͯ͠ΓɺߋʹϚοϋ਺

M ͸ϐτʔѹൺ^p_p⁰²

o ͷΈͷؔ਺ͱ͍ͯ͠Δɻ͜ͷఆٛΛ༻͍Ε͹ɺಈѹqɺҰ༷ྲྀ੩ѹp_∞ʹؔͯ͠΋ɺܭଌ

ྔͷ‘ਅ஋’ٴͼܭଌޡࠩΛ༻͍ͯɺ‘ਅ஋’ͱܭଌޡࠩͱʹҎԼͷ༷ʹ෼͚Δ͜ͱ͕Ͱ͖Δ:

q=q,0+δq

≈q,0+ ∂q

∂M(δM) + ∂q

∂po(δpo)

≈q,0+ ∂q

∂M dM d^p_p⁰²_o

δp02

po

+ ∂q

∂po

(δpo) p_∞=p_∞,0+δp_∞

≈p_∞,0+∂p_∞

∂M dM d^p_p⁰²_o

δp02

po

+∂p_∞

∂po

(δpo) δp02

po ≈ 1 po

δp02−p02

p²_o δpo

͜ΕΒ͔Βɺલ໘࣠ྗ܎਺CAF΁ͷܭଌޡࠩͷӨڹʹؔͯ͠ҎԼͷ༷ʹ੔ཧ͢Δ͜ͱ͕Ͱ͖Δ:

CAF =CAF,0+δCAF

=FA,0+δFA−Sb(p_∞,0+δp_∞−pb,0−δpb) (q,0+δq)S

≈CAF,0+δFA

qS +−Sb

qS (δp_∞−δpb)−CAFδq q

=CAF,0+δFA

qS + Sb

qSδpb− Sb

qS

∂p_∞

∂M +CAF

q

∂q

∂M dM

d^p_p⁰²_o

δp02

po

− Sb

qS

∂p_∞

∂po

+CAF

q

∂q

∂po

(δpo)

=CAF,0+AFδFA+Apbδpb+Ap02δp02

po +Apoδpo (1)

͜͜ͰɺCAF,0≡ ^FA,0−Sbq^(p,0^∞S^,0−pb,0) ͸લ໘࣠ྗ܎਺ͷ‘ਅ஋’Ͱ͋Γ·ͨɺ δp02

po ≈ 1

poδp02−p02

p²_o δpo

AF = 1 qS Apb = Sb

qS Ap02 =−

Sb

qS

∂p_∞

∂M +CAF

q

∂q

∂M dM

d^p_p⁰²_o Apo =−Sb

qS

∂p_∞

∂po −CAF

q

∂q

∂po

(2)

ͱ͢Δɻ্ࣜதͷ _p¹_o∂M^∂q ,_∂p^∂q_o,_p¹_o^∂p_∂M^∞,^∂p_∂p^∞_o,_d^dMp02

po ʹؔͯ͠͸ɺ׬શؾମؔ܎ࣜΑΓҎԼͷ༷ʹٻΊΔ͜ͱ͕Ͱ

͖Δ:

1 po

∂q

∂M =−γMM²−2 2

1 + γ−1 2 M²

−^2γ−1γ−1

∂q

∂po

=γM² 2

1 +γ−1 2 M²

−γ^γ−1

1 po

∂p_∞

∂M =−γM

1 + γ−1 2 M²

−^2γ−1γ−1

∂p_∞

∂po

=

1 + γ−1 2 M²

−γ−^γ1

d^p_p⁰²

o

dM = 4γ γ−1

M²

γ−1−2γM² + 1 (γ−1)M²+ 2

1 M

p02

po

લ໘࣠ྗ܎਺CAF ͷޡࠩ఻೻ͷࣜ(1)Ͱɺ࣠ྗ܎਺ਅ஋ʹ͓͚Δมಈͱ֤ܭଌޡࠩʹ͓͚Δมಈʹ͓͍ͯ૬ؔ

ؔ܎ͷٙΘΕΔ૊Έ߹Θͤ͸ɺϐτʔѹൺԾఆ஋^p_p⁰²_o ͱɺఽΈѹྗܭଌ஋poʹؔ͢ΔޡࠩͰ͋Γɺσʔλॲཧ ͷࡍʹϐτʔѹൺ^p_p⁰²_o Λ‘Ծఆ’͢ΔՕॴ͕ൃੜ͢Δ৔߹͕͋ΔɻͦͷΑ͏ͳ৔߹ɺͦͷԾఆͷͨΊʹੜ͡Δޡ

ࠩδ^p_p⁰²

o ͱɺϐτʔѹܭଌޡࠩδp02ɺఽΈѹܭଌޡࠩδpoͱΛ෼཭͠ɺಠཱͨ͠ޡࠩཁҼͱධՁ͢Δ΂͖ͱߟ͑

ΒΕΔɻ࣮ࡍɺଞͷδFA, δpb, δpoͳͲ͸ܭଌܥʹ͓͚ΔޡࠩͰ͋Γɺجຊతʹ͸ηϯαܥɾॲཧܥͷޡࠩͱ͠

ͯͦͷఔ౓͕૝ఆ͞ΕΔ΋ͷͰ͋Δ͕ɺಛʹδ^p_p⁰²_o ʹؔͯ͠͸ؾྲྀ෼෍΍࠶ݱੑ౳෩ಎݻ༗ͷཁૉ΋ؚ·ΕΔ͜

ͱʹ஫ҙ͕ඞཁͰ͋Δɻδ^p_p⁰²_o ͸෩ಎؾྲྀίΞதʹ͓͚Δ‘୅දతͳ’^p_p⁰²_o ͷ‘ܭଌޡࠩ’͋Δ͍͸ਪఆޡࠩͱ૝ఆ

͞ΕΔͱߟ͑Δ΂͖Ͱ͋Δ͕ɺҰൠʹ͸ͦͷΑ͏ͳ‘ܭଌ’͸ࠔ೉Ͱ͋ΔͨΊɺJAXAۃ௒Ի଎෩ಎʹ͓͍ͯै

དྷ͔ΒߦΘΕ͍ͯΔΑ͏ʹMach਺ʢ෼෍ʣ͸ෆมͰ͋Γϐτʔѹൺ΋·ͨෆมͰ͋ΔͱԾఆ͢Δ͜ͱ΍ɺ࣮

ଌ͞ΕΔϞχλʔ༻ϐτʔѹܭଌ͔Β‘୅දతͳ’^p_p⁰²_o Λਪఆ͢Δ͜ͱͳͲ͕ߟ͑ΒΕΔɻ͜͜ͰؾྲྀίΞ಺ͷ ఆΊΒΕͨҰ఺ʹ͓͍ͯϐτʔѹΛϞχλܭଌͦ͠ΕΛجʹ‘୅දతͳ’^p_p⁰²_o Λਪఆ͠ϐτʔѹൺΛٻΊΔ৔߹ɺ લड़ͷ௨Γδ^p_p⁰²_o ʹ͸ɺܭଌܥޡࠩͷଞɺϐτʔѹൺͷۭؒ෼෍ͷͨΊ‘୅දతͳ’p02͸ϐτʔѹϞχλʔܭଌ Ґஔʹ͓͚Δϐτʔѹp02sͱ͸ඞͣ͠΋Ұக͠ͳ͍͜ͱΛߟྀ͢Δඞཁ͕͋Δɻͦ͜Ͱɺ‘୅දత’ͳϐτʔѹ p02ΛɺϐτʔѹϞχλʔ’ܭଌ஋’p02sͱɺͦͷൺͱͯ͠ઃఆ͢Δ܎਺apʢap≡p02/p02s=ap+a^′_pʣͳͲʹ

෼͚ͯ੔ཧ͠ɺ෩ಎؾྲྀͷϐτʔѹ෼෍ޮՌΛධՁ͢Δ͜ͱͱ͢Δɻϐτʔѹൺͷ૬ରతۭؒ෼෍͕௨෩ຖʹ ෆมͰ͋Ε͹܎਺apͷมಈ෼a^′_p͸0ͱͳΔͱߟ͑ΒΕΔɻ͜ͷ༷ʹۭؒ෼෍ͷӨڹΛධՁ͠Α͏ͱͨ͠৔߹

‘୅දతͳ’ϐτʔѹൺ ^p_p⁰²_o ͸ϐτʔѹϞχλʔܭଌ஋^p_p^02s_o = ^p02s,0_po +δ^p_p^02s_o ౳Λ༻͍ͯɺ p02

po = p02

po

0

+δp02

po

≈ap

p02s

po

0

+δp02s

po

+a^′_p

p02s

po

0

≈ap

p02s

po

0

+apδp02s

po

+a^′_pp02s

po

ʹΑΓධՁͰ͖ɺߋʹp02,0≈app02s,0ͱԾఆ͢Ε͹ɺ^p_p⁰²

o‘ܭଌ’ʹ͓͚Δ‘ܭଌޡࠩ’δ^p_p⁰²

o ͸ɺ

δp02

po ≈apδp02s

po

+a^′_pp02s

po

(3) ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜Ͱap͸ؾྲֱྀਖ਼ࢼݧ࣌ͷฏۉ෼෍ΛجʹٻΊΔ͜ͱ͕Ͱ͖ɺa^′_p͸ؾྲֱྀਖ਼ࢼݧͷ ࡍʹऔಘ͞ΕͨಉҰஅ໘ʹ͓͚Δʢن֨Խͨ͠ʣϐτʔѹൺ෼෍ʹ͓͚Δ࠶ݱੑσʔλ͔Βਪఆ͢Δ͜ͱ͕Ͱ

͖Δ΋ͷͱߟ͑ΒΕΔɻ

ͱ͢Δɻ্ࣜதͷ _p¹_o∂M^∂q ,_∂p^∂q_o,_p¹_o^∂p_∂M^∞,^∂p_∂p^∞_o,_d^dMp02

po ʹؔͯ͠͸ɺ׬શؾମؔ܎ࣜΑΓҎԼͷ༷ʹٻΊΔ͜ͱ͕Ͱ

͖Δ:

1 po

∂q

∂M =−γMM²−2 2

1 + γ−1 2 M²

−^2γ−1γ−1

∂q

∂po

=γM² 2

1 +γ−1 2 M²

−γ^γ−1

1 po

∂p_∞

∂M =−γM

1 + γ−1 2 M²

−^2γ−1γ−1

∂p_∞

∂po

=

1 + γ−1 2 M²

−γ−^γ1

d^p_p⁰²

o

dM = 4γ γ−1

M²

γ−1−2γM² + 1 (γ−1)M²+ 2

1 M

p02

po

લ໘࣠ྗ܎਺CAFͷޡࠩ఻೻ͷࣜ(1)Ͱɺ࣠ྗ܎਺ਅ஋ʹ͓͚Δมಈͱ֤ܭଌޡࠩʹ͓͚Δมಈʹ͓͍ͯ૬ؔ

ؔ܎ͷٙΘΕΔ૊Έ߹Θͤ͸ɺϐτʔѹൺԾఆ஋^p_p⁰²_o ͱɺఽΈѹྗܭଌ஋poʹؔ͢ΔޡࠩͰ͋Γɺσʔλॲཧ ͷࡍʹϐτʔѹൺ^p_p⁰²_o Λ‘Ծఆ’͢ΔՕॴ͕ൃੜ͢Δ৔߹͕͋ΔɻͦͷΑ͏ͳ৔߹ɺͦͷԾఆͷͨΊʹੜ͡Δޡ

ࠩδ^p_p⁰²

o ͱɺϐτʔѹܭଌޡࠩδp02ɺఽΈѹܭଌޡࠩδpoͱΛ෼཭͠ɺಠཱͨ͠ޡࠩཁҼͱධՁ͢Δ΂͖ͱߟ͑

ΒΕΔɻ࣮ࡍɺଞͷδFA, δpb, δpoͳͲ͸ܭଌܥʹ͓͚ΔޡࠩͰ͋Γɺجຊతʹ͸ηϯαܥɾॲཧܥͷޡࠩͱ͠

ͯͦͷఔ౓͕૝ఆ͞ΕΔ΋ͷͰ͋Δ͕ɺಛʹδ^p_p⁰²_o ʹؔͯ͠͸ؾྲྀ෼෍΍࠶ݱੑ౳෩ಎݻ༗ͷཁૉ΋ؚ·ΕΔ͜

ͱʹ஫ҙ͕ඞཁͰ͋Δɻδ^p_p⁰²_o ͸෩ಎؾྲྀίΞதʹ͓͚Δ‘୅දతͳ’^p_p⁰²_o ͷ‘ܭଌޡࠩ’͋Δ͍͸ਪఆޡࠩͱ૝ఆ

͞ΕΔͱߟ͑Δ΂͖Ͱ͋Δ͕ɺҰൠʹ͸ͦͷΑ͏ͳ‘ܭଌ’͸ࠔ೉Ͱ͋ΔͨΊɺJAXAۃ௒Ի଎෩ಎʹ͓͍ͯै

དྷ͔ΒߦΘΕ͍ͯΔΑ͏ʹMach਺ʢ෼෍ʣ͸ෆมͰ͋Γϐτʔѹൺ΋·ͨෆมͰ͋ΔͱԾఆ͢Δ͜ͱ΍ɺ࣮

ଌ͞ΕΔϞχλʔ༻ϐτʔѹܭଌ͔Β‘୅දతͳ’^p_p⁰²_o Λਪఆ͢Δ͜ͱͳͲ͕ߟ͑ΒΕΔɻ͜͜ͰؾྲྀίΞ಺ͷ ఆΊΒΕͨҰ఺ʹ͓͍ͯϐτʔѹΛϞχλܭଌͦ͠ΕΛجʹ‘୅දతͳ’^p_p⁰²_o Λਪఆ͠ϐτʔѹൺΛٻΊΔ৔߹ɺ લड़ͷ௨Γδ^p_p⁰²_o ʹ͸ɺܭଌܥޡࠩͷଞɺϐτʔѹൺͷۭؒ෼෍ͷͨΊ‘୅දతͳ’p02͸ϐτʔѹϞχλʔܭଌ Ґஔʹ͓͚Δϐτʔѹp02sͱ͸ඞͣ͠΋Ұக͠ͳ͍͜ͱΛߟྀ͢Δඞཁ͕͋Δɻͦ͜Ͱɺ‘୅දత’ͳϐτʔѹ p02ΛɺϐτʔѹϞχλʔ’ܭଌ஋’p02sͱɺͦͷൺͱͯ͠ઃఆ͢Δ܎਺apʢap ≡p02/p02s=ap+a^′_pʣͳͲʹ

෼͚ͯ੔ཧ͠ɺ෩ಎؾྲྀͷϐτʔѹ෼෍ޮՌΛධՁ͢Δ͜ͱͱ͢Δɻϐτʔѹൺͷ૬ରతۭؒ෼෍͕௨෩ຖʹ ෆมͰ͋Ε͹܎਺apͷมಈ෼a^′_p͸0ͱͳΔͱߟ͑ΒΕΔɻ͜ͷ༷ʹۭؒ෼෍ͷӨڹΛධՁ͠Α͏ͱͨ͠৔߹

‘୅දతͳ’ϐτʔѹൺ^p_p⁰²_o ͸ϐτʔѹϞχλʔܭଌ஋^p_p^02s_o = ^p02s,0_po +δ^p_p^02s_o ౳Λ༻͍ͯɺ p02

po = p02

po

0

+δp02

po

≈ap

p02s

po

0

+δp02s

po

+a^′_p

p02s

po

0

≈ap

p02s

po

0

+apδp02s

po

+a^′_pp02s

po

ʹΑΓධՁͰ͖ɺߋʹp02,0≈app02s,0ͱԾఆ͢Ε͹ɺ^p_p⁰²

o‘ܭଌ’ʹ͓͚Δ‘ܭଌޡࠩ’δ^p_p⁰²

o ͸ɺ

δp02

po ≈apδp02s

po

+a^′_pp02s

po

(3) ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜Ͱap͸ؾྲֱྀਖ਼ࢼݧ࣌ͷฏۉ෼෍ΛجʹٻΊΔ͜ͱ͕Ͱ͖ɺa^′_p͸ؾྲֱྀਖ਼ࢼݧͷ ࡍʹऔಘ͞ΕͨಉҰஅ໘ʹ͓͚Δʢن֨Խͨ͠ʣϐτʔѹൺ෼෍ʹ͓͚Δ࠶ݱੑσʔλ͔Βਪఆ͢Δ͜ͱ͕Ͱ

͖Δ΋ͷͱߟ͑ΒΕΔɻ

ߋʹϐτʔѹϞχλܭଌྔp02sͷޡࠩͷධՁʹؔͯ͠ɺϐτʔѹൺͷ‘ܭଌ஋’͸ϐτʔѹൺͷਅ஋

p02s

po

oΛ

༻͍ͯɺ

p02s

po = p02s

po

o

+ 1

poδp02s−p02s

p²_o δpo

ͱॻ͚Δ͸ͣͰ͋Δɻ͜͜ͰɺҰ༷ྲྀMach਺͕ෆมͰ͋ΔͱԾఆ͢Δैདྷͷํ๏ʹ͓͍ͯ͸΋ͪΖΜɺϐτʔ ѹϞχλʕܭଌʹΑΓ֤௨෩ʹ͓͍ͯҰ༷ྲྀMach਺ΛٻΊΑ͏ͱ͢Δ৽ͨͳํ๏ʹ͓͍ͯ΋ɺϐτʔѹܭଌ

࣌ʹ͓͚Δଌఆ஋͕௨෩தҰఆͰ͋ΔͱԾఆ͢ΔҎ্ɺϐτʔѹൺԾఆྔ͕ଘࡏ͢ΔɻͦͷϐτʔѹൺԾఆʹ ΑΔ‘ޡࠩ’Λݟੵ΋Δඞཁ͕͋ΓɺԾఆ͍ͯ͠Δϐτʔѹൺෆม஋Λ_p

02s

po

Aͱ͢Ε͹ɺ͜ͷԾఆྔͰ͋Δ _p

02s

po

Aͱ‘ਅ஋’_p

02s

po

oͱͷ͕ࠩ͜ͷ৔߹ධՁ͢Δ΂͖‘ޡࠩ’Ͱ͋Δͱߟ͑ΒΕΔͷͰɺ δp02s

po = p02s

po

A

− p02s

po

0

= p02s

po

A−p02s

po

+ 1 po

δp02s−p02s

p²_o δpo

ͦͷͨΊɺϐτʔѹൺԾఆʹ൐͏ޡࠩཁҼͱͯ͠

δ^∗p02s

po ≡ p02s

po

A−p02s

po

(4) Λఆٛ͢Ε͹ɺ‘୅දతͳ’ϐτʔѹൺޡࠩδ^p_p⁰²_o ͷࣜ(3)͸ɺ

δp02

po ≈apδ^∗p02s

po

+ap

po

δp02s−p02sap

p²_o δpo+a^′_pp02s

po

(5) Ͱ͋ΔɻͦͷͨΊɺࣜ(1)ͱ૊Έ߹ΘͤΔ͜ͱͰɺ

CAF ≈CAF,0+AFδFA+Apbδpb+apAp02δ^∗p02s

po +Ap02

ap

poδp02s

+

Apo−Ap02

p02sap

p²_o

δpo+a^′_p p02s

po

Ap02

(6)

͜͜Ͱ֤߲ಉ࢜ͷ૬ؔ͸ͳ͍΋ͷͱߟ͑Δ͜ͱ͕Ͱ͖ΔΑ͏ʹͳͬͨͨΊɺલ໘࣠ྗ܎਺‘ܭଌ݁Ռ’ͷฏۉ஋

ٴͼඪ४ภࠩσCAF ͸ɺ

CAF ≈CAF,0+AFδFA+Apbδpb+Ap02apδ^∗p02s

po +Ap02

ap

poδp02s+

Apo −Ap02

p02sap

p²_o

δpo (7) σ²_CAF ≈σ²_CAF,0+A²_Fσ²_δFA+A²_pbσ²_δpb

+ (Ap02ap)²σ²_δ∗p02s po +

Ap02

ap

po

2

σ_δp²_02s+

Apo−Ap02

p02sap

p²_o 2

σ_δp²_o+ p02s

po

2

A²_p02σ_a²_p (8)

ͱද͢͜ͱ͕Ͱ͖Δɻࣜ(7),(8)ͰδCAF =CAF−CAF,0͸͍ΘΏΔจݙ[6]Ͱ੔ཧ͞ΕΔbiasޡࠩʹ૬౰͠ɺ δ^′_CAF ͸precisionޡࠩʹ૬౰͢Δͱߟ͑ΒΕΔɻ͜͜ͰσCAF,0͸࣠ྗ܎਺ਅ஋ͷมಈΛҙຯ͓ͯ͠Γɺઌʹ

‘ఆٛ’͞ΕͨM, q, p_∞͕෺ཧతʹఆٛ͞ΕΔMach਺ɺಈѹɺ੩ѹͱҰக͠ɺ·ͨؾྲྀ͕Ұ༷Ͱ͋Γɺ͔ͭRe

਺ɺM ਺͕େ͖͘มԽ͠ͳ͍ݶΓ͸ຊདྷ͸0ͱͳΔͱظ଴͞ΕΔྔͰ͋ΔɻͦͷͨΊɺKΛแׅ౓܎਺ɺKB

ΛbiasޡࠩʹରԠ͢Δแׅ౓܎਺ʹ૬౰͢Δ܎਺ͱͯ͠ɺ෩ಎࢼݧʹ͓͚Δෆ͔֬͞ʢbiasޡ֤߲ࠩͷແ૬ؔ

ΛԾఆͯ͠ʣ K_B²

CAF−CAF,0²

+K²σ_C²_AF ͸σCAF,0= 0ΛԾఆͯ͠ҎԼͷ௨ΓධՁ͢Δ͜ͱ͕Ͱ͖Δɿ U_C²_AF ≈A²_FU_δF² _A+A²_pbU_δp²_b+ (Ap02ap)²U_δ²_∗^p02s

po +

Ap02

ap

po

2

U_δp²_02s

+

Apo−Ap02

p02sap

p²_o 2

U_δp²_o+ p02s

po

2

A²_p02U_a²_p

(9)

͜͜Ͱɺ

U_δF²_A =K_B²δFA

2+K²σ²_δFA, U_δp²_b =K_B²δpb

2+K²σ_δp²_b, U_δ²_∗^p02s

po =K_B²δ^∗p02s

po 2

+K²σ_δ²_∗^p02s po

U_δp²_02s =K_B²δp02s

2+K²σ_δp²_02s, U_δp²_o =K_B²δpo

2+K²σ²_δpo, U_a²_p=K²σ²_ap

(10)

ͱ͢Δɻ

͜͜ͰɺҰ༷ྲྀMach਺ΛࢼݧΩϟϯϖʔϯɺRunʹ߆ΘΒͣৗʹෆมͰ͋ΔͱԾఆ͢Δ৔߹ͱɺ௨෩ຖʹϐ τʔѹϞχλʕܭଌʹΑΓҰ༷ྲྀMach਺Λ౎౓ਪఆ͢Δ৔߹ͱͰͷޡࠩධՁͷҧ͍͸ɺࣜ(4)ͷධՁͷҧ͍

ʹΑΓੜ͡ΔɻৗʹҰ༷ྲྀMach਺ΛෆมͱԾఆ͢Δ৔߹͸ɺԾఆ஋_p

02s

po

A͸୯Ұͷෆมྔͱ͠ɺࢼݧΩϟ ϯϖʔϯɾ௨෩RunΛލ͍ͰͦͷԾఆ஋͔Βͷࠩͷฏۉ஋ɾඪ४ภࠩΛٻΊΔඞཁ͕͋Δͷʹର͠ɺ௨෩ຖʹ ϐτʔѹϞχλʕܭଌʹΑΓҰ༷ྲྀMach਺Λ௨෩ຖʹਪఆ͢Δख๏ʹ͓͍ͯ͸ɺ௨෩ຖʹઃఆ͞ΕͨԾఆ஋

_p

02s

po

Aͱͷ͕ࠩ௨෩தʹͲͷఔ౓มԽ͠͏Δ͔ΛධՁ͢Δඞཁ͕͋Δɻ

2.2 ؾྲྀͷඇҰ༷ੑ͕ٴ΅͢ 6 ෼ྗܭଌ΁ͷӨڹ

ؾྲྀʹඇҰ༷ੑ͕͋Δ৔߹ɺ‘୅දతͳ’p02Λਪఆ͢Δࡍͷޡࠩ఻೻ͷධՁΛલઅͰݕ౼͕ͨ͠ɺඇҰ༷ͳؾྲྀ

ʹΑΔۭؾྗͷมಈ͸ɺલઅͷ੔ཧͰ͸ۭྗ܎਺‘ਅ஋’(CAF,0)ͷมಈͱͯ͋͠ΒΘΕΔ͜ͱʹͳΔɻ͔͠͠

Ұൠʹ෩ಎࢼݧʹΑΓಘΒΕΔۭྗಛੑσʔλ͸ɺҰ༷ͳؾྲྀதʹ͓͍ͯൃੜ͢ΔۭؾྗΛਪఆ͢ΔͨΊͷ΋

ͷͰ͋Δ͜ͱΛ೦಄ʹஔ͘ͱɺؾྲྀͷඇҰ༷ੑʹΑΓੜ͍ͯ͡ΔՄೳੑͷ͋ΔྔΛਪఆ͠໌ࣔ͢Δ͜ͱ͕༗ӹ Ͱ͋Δͱߟ͑ΒΕΔɻͦͷඇҰ༷ੑͷޮՌ͸ࢼݧ໛ܕҐஔΛ෩ಎؾྲྀதʹ͘·ͳ͘มԽͤͨ͞ͱ͖ʹಘΒΕΔ

ۭྗྗɾϞʔϝϯτͷมಈ෼ͱͯ͋͠ΒΘΕΔ΋ͷͱߟ͑Δ΂͖Ͱ͋Δɻͦ͜Ͱɺ௕͞εέʔϧ͕LdͰ͋Δ Α͏ͳඇҰ༷ੑͷɺ௕͞εέʔϧ͕LmͰ͋Δ໛ܕ্ʹಇۭ͘ؾྗ΁ͷޮՌͷݟੵ΋ΓΛɺہॴಈѹqfͱہ ॴ܎਺Cpٴͼද໘ੵdsͱͷੵͰہॴۭؾྗ͕ද͞ΕΔͱ͍͏ϞσϧԽʹΑΓߦ͏͜ͱͱͨ͠ɻ͜ͷہॴ܎਺

͸ɺද໘ѹྗ͕ಈѹʹൺྫ͢Δͱ͍͏Ծఆʹجͮ͘΋ͷͰ͋Δ͕ɺಈѹʹൺ΂Ұ༷ྲྀ੩ѹ͕ۃ୺ʹখ͘͞ͳΔ ۃ௒Ի଎ྲྀʹ͓͍ͯ͸ۙࣅతʹѹྗ܎਺ͱ߹க͢Δ΋ͷͰ͋Γɺͦͷൣғʹ͓͍ͯ͸Ծఆͷଥ౰ੑ͸֬ೝͰ͖

Δɻ·ͣہॴಈѹΛܭଌྔͰ͋Δϐτʔѹྗ෼෍͔Βਪఆ͢Δඞཁ͕͋Δ͕ɺ͜͜Ͱ΋ۃ௒Ի଎ྲྀͷ༷ʹൺֱ

తߴMach਺Ͱ͸ɺϐτʔѹͷMach਺ґଘੑ͕খ͘͞ͳΔ͜ͱ͔Βɺϐτʔѹͱಈѹͷൺ͸΄΅Ұఆ஋ͱۙ

ࣅͰ͖Δ:

p02

q = p02

po

po

p_∞ 2

γM² →2γ^−γ−1γ

γ+ 1 2

^γ+1γ−1

≈1.84 (M → ∞) (11)

͜ΕʹΑΓɺ෩ಎֱਖ਼ࢼݧͰಘΒΕ͍ͯΔϐτʔѹൺ෼෍͔Βಈѹ෼෍Λਪఆ͢Δ͜ͱ͕Ͱ͖Δɻ

͜͜Ͱɺզʑͷڵຯͷର৅ͱ͢Δ΋ͷͱ͸ɺྫ͑͹ؾྲྀίΞྖҬ಺ͷҰ఺xoͱ໛ܕ࠲ඪݪ఺͕Ұக͢Δࢼݧ

໛ܕʹಇۭ͘ؾྗΛߟ͑ͨͱ͖ʹɺxoͷมԽͱͱ΋ʹͦͷ໛ܕʹՃΘΔۭؾྗf(xo)ͷมԽ͢ΔྔͰ͋Γɺͦ

Ε͸

σ²_f ≈(i,j,k) 1 Vo

Vo





fx(xo)−fx

2

fy(xo)−fx² fz(xo)−fx²



dvxo

≈(i,j,k) 1 Vo

αo̸=0





|Fx(αo)|²

|Fy(αo)|²

|Fz(αo)|²



dαo

2π d

βo

2π

dγo

2π

≈(i,j,k)

k,l,m







 N^F1^xk,l,mN2N3

2

N^F1^yk,l,mN2N3

²

N^F1^zk,l,mN2N3

²







