In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
Basic properties concerning Earth's orbit about the Sun and some of the physics parameters involved in the problem.
In [16]:
d_perihelion = 1.47098074e11 # m
d_aphelion = 1.52097701e11 # m
d_average = 1.4959787e11 # m
GMsun = 1.3271244002e20 # m^3 s^-2
MEarth = 5.9722e24 # kg
c = 2.99792458e8 # m s^-1
Based on these numbers, we can estimate the gravitational force at aphelion and perihelion to be
In [17]:
g_perihelion = GMsun*MEarth/np.power(d_perihelion, 2)
g_aphelion = GMsun*MEarth/np.power(d_aphelion, 2)
g_average = GMsun*MEarth/np.power(d_average, 2)
Alternatively, we can base the estimate on the relative strength, which relieves uncertainties due to GMsun and MEarth.
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g_ratio = np.power(d_aphelion/d_perihelion, 2)
We see that the strength of gravitational force between the Earth and Sun is about 7% larger at perihelion as compared to aphelion.
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print g_ratio
Now we must convert the relative difference in gravitational force into a difference in time dilation for various processes. Since we're dealing with the Sun, we may adopt the Schwarzschild metric, which suggests \begin{equation} t_0 = t_f \left( 1 - \frac{r_s}{r} \right)^{1/2}. \end{equation} The Schwarzschild radius for the Sun is
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r_s = 2.0*GMsun/np.power(c, 2)
which allows us to compute the relative difference in $t_0$ based on the variable Earth-Sun distance.
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t0_ratio = np.sqrt(1.0 - (r_s / d_perihelion)) / np.sqrt(1.0 - (r_s / d_aphelion))
t0_ratio_aphelion = np.sqrt(1.0 - r_s / d_aphelion) / np.sqrt(1.0 - r_s / d_average)
t0_ratio_perihelion = np.sqrt(1.0 - r_s / d_perihelion) / np.sqrt(1.0 - r_s / d_average)
The ratio is
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print t0_ratio
which is small, but perhaps non-neligible when it comes to seasonal variations in the decay of Potassium-40, for example. In the case of the DAMA experiment, they have 250 kg of sodium iodide crystals with a small 20 ppb contamination of Potassium-40. Potassium-40 is radioactive with a long half-life of $1.251\times10^9$ years. There are two primary decay channels: \begin{equation} ^{40}\textrm{K}\ \rightarrow\ ^{40}\textrm{Ca} + \beta^{-} + \bar{\nu} \end{equation} and \begin{equation} ^{40}\textrm{K} + \beta^{-}\ \rightarrow\ ^{40}\textrm{Ar} + \gamma + \nu \end{equation} which occur about 90% and 10% of the time, respectively.
The molar mass of sodium iodide is 149.89 g mol$^{-1}$, meaning in 250 kg there are
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moles_NaI = 250.0e3/149.89
print moles_NaI, " moles of NaI,"
which translates into
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N_NaI = moles_NaI*6.022140857e23
print N_NaI, "NaI molecules."
Assuming the maximum contamination fraction for Potassium-40 of 20 ppb, there are roughly
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N_40K = 20.0*N_NaI/1.0e9
print N_40K, "Potassium-40 atoms in the sample."
Now our task is to estimate the approximate increase or decrease in Potassium-40 decays based on variations in the Sun-Earth gravitational potential.
During a single half-life for Potassium-40, approximately $1\times10^{19}$ atoms decay. Given a mean half-life of $1.251\times10^{9}$ years, there will be approximately
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decay_rate_average = 1.0e19/(1.251e9*60.0*60.0*24.0*365.256366)
print decay_rate_average, "decays per second on average."
The decay rate at perihelion and aphelion will be slightly different due to the gravitational potential of the Earth.
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decay_rate_aphelion = 1.0e19/(1.251e9*60.0*60.0*24.0*365.256366*t0_ratio_aphelion)
print decay_rate_aphelion, "decays per second at aphelion."
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decay_rate_perihelion = 1.0e19/(1.251e9*60.0*60.0*24.0*365.256366*t0_ratio_perihelion)
print decay_rate_perihelion, "decays per second at perihelion."
Thus, during a given 24 hour period, one would expect
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decay_rate_aphelion = 1.0e19/(1.251e9*365.256366*t0_ratio_aphelion)
print decay_rate_aphelion, "decays per day at aphelion and"
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decay_rate_perihelion = 1.0e19/(1.251e9*365.256366*t0_ratio_perihelion)
print decay_rate_perihelion, "decays per day at perihelion."
which yields a difference of
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print decay_rate_perihelion - decay_rate_aphelion, "decays per day between summer and winter."
which leads to a difference in
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print (decay_rate_perihelion - decay_rate_aphelion)/250.0, "decays per kg per day."
This level is far below the variation in counts per day observed by the DAMA experiment, which is approximately 0.05 cpd/kg/keV.