我們使用中子星冷卻的實驗數據來預測軸子質量的範圍。首先，我們簡
要討論了中子星的比熱與溫度之間的關係是如何推導出來的。接著，我們
將概述與 URCA 過程中中子產生相關聯的輻射強度的推導，該過程導致熱
能的損失。接著，我們納入軸子的影響來增強熱能的散失，以期接近實驗
結果。這將使我們能夠推斷軸子與中子之間的耦合常數，以及在兩個標準
偏差內可能的軸子質量範圍。我們在附錄中包含了 Python 程式碼供參考。

We use the experimental data of neutron star cooling to predict the
range of axion mass. Beginning by briefly discussing how the relationship
between specific heat and temperature of a neutron star is derived. We will
then outline the derivation of the luminosity associated with the production
of neutrons in the URCA process, which leads to the loss of thermal energy.
Next, we incorporate the influence of axions to enhance the heat dissipation,
aiming to approach experimental results. This will allow us to infer the
coupling constants between axions and neutrons, as well as the possible
range of axion masses within two standard deviations. We include the
Python codes in the appendix for reference.

Contents
1 Introduction 6
2 The cooling mechanism 6
2.1 The Heat Capacity . . . . . . . . . . . . . . . . . . . . 6
3 A completely degenerate electron gas 9
4 fields 19
5 The scattering amplitude 20
6 The URCA process 22
6.1 The Modified URCA Rate . . . . . . . . . . . . . . 26
7 Neutron Star Cooling 29
8 Axions 31
8.1 Axion Types . . . . . . . . . . . . . . . . . . . . . . 31
9 Neutron Star Cooling With Axions 33
10 the Results 38
11 Summary 43
A the Python codes 44
B source of some constants 51

References
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Benjamin R Safdi. Upper limit on the qcd axion mass from isolated neutron
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