Evidence for the existence of Planet 9 has been accumulating for about 20 years, [36] [37] [38] [39] [40]. Analysis of the evidence, the anomalous orbits of some Kuiper Belt objects, suggests the existence of a new planet of mass ~7 Earth masses, in an orbit of eccentricity ~0.3 with a semi major axis of ~380 AU inclined at ~15 degrees to the ecliptic plane [1]. However, the parameter estimates, based just on the anomalous clustering of a few distant objects, remain very uncertain [1].

Planet heliographic longitudes on January 01, 1965, were obtained from https://omniweb.gsfc.nasa.gov/coho/helios/heli.html. Group sunspot numbers 1610 to 2015 [41], were obtained from https://svalgaard.leif.org/research/gn-data.htm. Reconstructed SSN, −6755 to 1885, was obtained from https://www2.mps.mpg.de/projects/sun-climate/data/SN_composite.txt.

In view of the limited accuracy of the orbital parameters of Planet 9 SIM is calculated using a model where all the planets move about the Sun in circular orbits in the ecliptic plane. This is a good approximation for the known planets that have a significant effect on SIM, i.e. Jupiter, Saturn, Uranus and Neptune, as the orbits these planets are nearly circular and have low inclination to the ecliptic plane. As shown in Figure 1, the SIM, calculated using circular orbits for the eight known planets, is scarcely distinguishable from SIM calculated using exact planet orbits based on ephemeris data [6] [32] [34] [42]. However using a circular orbit for Planet 9 is an approximation as its orbit is projected to be eccentric, ε ~ 0.3, and inclined to the ecliptic, i ~ 15˚ [1]. Including eccentricity and inclination of Planet 9 in the calculation of SIM would greatly increase the complexity of the calculation [42], and is outside the scope of this paper. Spectral analysis will be the major investigative tool in this paper and for basic spectral analysis like FFT equal time intervals are essential. With circular orbits, as used in this paper, equal time intervals occur naturally in the calculation of SIM whereas for eccentric orbits equal angular intervals occur naturally and the conversion to the equal time intervals required for FFT is complex. The orbital parameters of the planets used in the calculation are given in Table 1.

The orbital period of Planet 9, T₉ = 7400 years, is obtained from the value of the semi-major axis, a₉ = 380 AU [1] and the use of Kepler’s 3rd Law, T²/a³ = constant. If T is measured in years and a is measured in AU, the constant = 1.

The time variation of the coordinates, (x_i,y_i), of the ith planet relative to the Sun as origin at (0,0) are calculated using

x i = r i cos ( ω i t + φ i ) y i = r i sin ( ω i t + φ i ) (1)

where r_i is the orbital radius of the ith planet, the angular frequency ω_i = 2π/T_i, the phase angle in radians, ϕ_i = (π/180)L_i, and L_i is the heliographic inertial longitude of the planet in degrees on January 01, 1965. The coordinates, (x_PCM, y_PCM), of the planetary centre of mass (PCM) relative to the Sun are given by

x P C M = ∑ m i x i / ∑ m i y P C M = ∑ m i y i / ∑ m i (2)

The distance between the Sun and the planetary centre of mass, r_PCM, is given by

r P C M = ( x P C M 2 + y P C M 2 ) 1 / 2 (3)

The distance between the Sun and the barycentre, R_B, is

R B = r P C M [ ∑ m i / ( ∑ m i + m S U N ) ] (4)

The Sun to barycentre distance is usually expressed as the ratio R_B/R_SUN where R_SUN is the radius of the Sun, 0.0046 AU.

Figure 1(a) shows the time variation of R_B/R_SUN for the eight planet system, m₉ = 0, and Figure 1(b) shows R_B/R_SUN for the nine planet system when m₉ = 7m_E. The most striking feature of Figure 1 is that the frequency of the cycles in R_B/R_SUN has doubled when m₉ = 7m_E. A second feature is that there is a significant centennial scale variation, ~170 year in period, when m₉ = 7m_E that is not apparent when m₉ = 0.

Figure 2 compares the spectral content of the two variations of R_B/R_SUN in Figure 1. The spectrum of R_B/R_SUN for the eight planet system, m₉ = 0, is dominated by a component at period 19.5 years, with weaker components at 12.6 and 13.7 years and minor components at 35 and 46 years. Thus the periodicity of the eight planet variation is primarily bi-decadal. The periodicity of R_B/R_SUN for the nine planet system is dominated by the component at period 11.9 years with weaker components at 8.4, 13.7, 29.6, ~86 and ~172 years. Thus the periodicity of R_B/R_SUN for the nine planet system is primarily decadal. The component at 11.9 years is the synodic period of Jupiter with Planet 9. Similarly the components at 29.6 years, ~86 years and ~172 years are due to synodic periods with

Saturn, Uranus and Neptune respectively. The synodic period of a component is given by T = 1/(1/T_P − 1/T₉).

A feature of R_B/R_SUN for the nine planet system is the presence of moderately strong components at ~86 years and ~172 years. The presence of these longer period components is also clearly evident in the time variation of Figure 1(b) where minima in the long term average value of R_B/R_SUN occur at ~170 year intervals with the minima separated by broad maxima. The pattern of broader maxima alternating with sharper minima is the result of the interference of the ~86 year and ~172 year cycles. That is, the deep centennial scale minima in Figure 1(b) occur when the two cycles are both in the negative part of their cycles. The labels Maunder, Dalton and Modern in Figure 1(b) correspond approximately to the central times of grand solar minima in SSN and the times clearly align with the centennial scale minima in R_B/R_SUN.

The transition from the primarily bi-decadal SIM for the eight planet system to a primarily decadal SIM for the nine planet system is quite dramatic so it is interesting to follow how this comes about. Figure 3 compares the orbits of the Sun about the barycentre for the eight planet and the nine planet systems during the interval 1890 to 1913, centred on the year 1900, see Figure 1(a) and Figure 1(b). In the left hand diagram of Figure 3 the first minimum in R_B/R_SUN occurs at 1 followed by a maximum at 2 and another minimum at 3, with about 20 years between minima, i.e. ~bi-decadal periodicity. The right hand diagram shows the Sun orbit about the barycentre when m₉ = 7m_E for two values of Planet 9 longitude, L₉ = 60˚ and L₉ = 240˚. The heliographic inertial longitude, L, is measured anticlockwise from the positive x axis.

For the L₉ = 60˚ case there is a first minimum at 1, a first maximum at 2, a second minimum at 3, a second maximum at 4, and a third minimum at 5, with about 10 years between minima, i.e. ~decadal periodicity. The phase of the

~decadal cycle is sensitive to Planet 9 longitude. This is illustrated in Figure 3 for the case of L₉ = 240˚ where the orbital pattern due primarily to Jupiter, Saturn, Uranus, and Neptune has moved from the upper right hand quadrant to the lower left hand quadrant of the diagram. The minimum in R_B/R_SUN at 3 occurring at about 1901 is now a maximum in R_B/R_SUN at 3# occurring at about 1901. It is apparent that when the longitude of Planet 9 changes by 180˚ the phase of the short, ~decadal, cycle in R_B/R_SUN also changes by 180˚.

The group sunspot number (GSS) [41], extends from 1610 to 2015 (Figure 4). By removing the 20 year running average we obtain the primarily decadal variation of GSS. The GSS, before 1700, during the Maunder Minimum is of low accuracy [41], so here we use GSS in the interval between 1700 and 2015 to compare with the variation of R_B/R_SUN (Figure 5). In Figure 5 we use the normalised variations, i.e. the variations divided by their standard deviation, to compare the GSS and R_B/R_SUN. There are 28 GSS cycles between 1705.5 and 2014.5. Thus, the average GSS period between the maximum at 1705.5 and the maximum at 2014.5 is 309/28 = 11.03 years. If the hypothesised forcing of SSN due to SIM occurs at the dominant periodicity of R_B/R_SUN in the decadal time range, i.e. at 11.9 years, see

Figure 2, we would expect 309/11.9 = 26 forcing cycles during this interval. That is, there are 28 GSS cycles compared with 26 SIM cycles during this 309 year long interval. As a result there are two 360˚ phase jumps apparent in Figure 5. The first occurs in the interval between 1729 and 1788, (at ~1758), and the second occurs between 1907 and 1980, (at ~1943). Stephani et al. [43] observed phase jumps in the Schwabe cycle at around the same times as observed here.

As is evident from Figure 3, the phase of the dominant decadal periodicity in R_B/R_SUN is strongly dependent on the longitude of Planet 9. For example, the minimum labelled 3 in Figure 3 of the orbital pattern when L₉ = 60˚ becomes a maximum, labelled 3#, in the orbital pattern when L₉ = 240˚. If the correlation coefficient between GSS variation and the R_B/R_SUN variation is calculated as the Planet 9 longitude is varied the correlation passes through a sharp maximum of 0.3 when L₉ = 60˚ as shown in Figure 6. The moderately high correlation coefficient is a result of the fairly close correspondence of the dominant periodicity in R_B/R_SUN, 11.9 years, and the average periodicity of the GSS between 1700 and 2015, 11.03 years. When the correlation coefficient between GSS and R_B/R_SUN for the eight planet system is found by a similar analysis for the same time interval the correlation coefficient is 0.034. A heliographic inertial longitude of L₉ = 60˚ locates Planet 9, at the present time, in the vicinity of the constellation Leo, near right ascension ~11 hours, declination ~+15˚.

The variability of solar activity over timescales longer than several centuries can

only be studied by proxy records of solar activity derived from measurements of the concentration of the cosmogenic isotopes, radiocarbon ¹⁴C in tree trunks and ¹⁰Be in polar ice, that accumulated due to the effect of cosmic rays [22] [52] [53] [54] [55]. Due to the inverse relationship between cosmic rays and solar activity it was possible to reconstruct the variation of SSN over the approximately 9,000 year Holocene interval, −7000 BC to 2000 AD [22] [56]. Analysis of the record shows a variation, nominally the Hallstatt cycle, of periodicity about 2400 years. The Hallstatt cycle and a longer period cycle were also observed in the ¹⁴C record [13]. Usoskin et al. [22], concluded that the ~2400 year Hallstatt cycle is most likely a property of long term solar activity. They show, by superposed epoch analysis that grand minima in SSN tend to cluster at minima of the ~2400 year cycle. The ~2400, ~1000, ~85, ~60 and ~30 year period components are important components in solar activity, cosmic ray flux and cosmogenic series and have been extensively reported [13] [20] [22] [55] [57] [58]. In this section we show that the spectral content of R_B/R_SUN, calculated when Planet 9 is included, exhibits the Hallstatt ~2400 year cycle, the Gleissberg ~88 year cycle, the ~60 year cycle, and the ~30 year cycle, Figure 9(a). Conversely, in the spectrum of R_B/R_SUN, calculated without Planet 9, these long term components of reconstructed solar activity are insignificant, Figure 9(b).

Figure 9(a) and Figure 9(b) compare the spectral content of R_B/R_SUN for the eight planet and nine planet systems when assessed between year −8000 and year +8000. Three overlapping nine planet spectra are shown in Figure 9(a). Each spectrum is for a different value of semi-major axis and the corresponding Planet 9 orbital period. Clearly, the low frequency components of R_B/R_SUN are sensitive to changes in Planet 9 semi-major axis and period. Therefore the variation of Planet 9 period provides a means to fit the lowest frequency component of R_B/R_SUN to the previously observed ~2400 year period of the Hallstaat cycle in reconstructed SSN by varying the orbital radius or orbital period of Planet 9. Figure 9(a) indicates that a close fit is obtained with a₉ ~ 366 AU, or equivalently, since T and a are related, T₉ ~ 7000 years. The value a₉ ~ 366 AU is close to the best estimate value, a₉ = 380 AU obtained by [1].

Figure 10 compares the eight planet and the nine planet time variations of R_B/R_SUN from year −8000, (8000 BC) to year +8000, (8000 AD). The points are calculated at 2400 day intervals so that the label S5 indicates a running average

over 5 points or a 12,000 day, 33 year interval and the label S100 indicates a running average over a 660 year interval. There is a very strong centennial time range cycle of ~170 year period and a weaker millennial time range cycle of ~2400 year period. Comparing Figure 10(a) where L = 60˚ and Figure 10(b) where L = 240˚ indicates that the ~2400 year cycle is sensitive to change in Planet 9 longitude, i.e. the phase of the ~2400 year cycle shifts by 180˚ when the longitude of Planet 9 shifts by 180˚.

The type of variation in Figure 10 arises when the major components, in this case the ~176 year and ~88 year period components are not exactly harmonically related. If the two components are harmonically related as suggested by the ratio 176/88 = 2.0 then a stationary pattern would result rather than the slowly varying pattern in Figure 10. Assuming the longer period component derives from the Uranus-Neptune conjunction period of 172 years we expect the synoptic period of the conjunction Uranus/Neptune/Planet 9 (period ~7000 years) to be 176.5 years and for the synoptic period of the harmonic of 172 years (86 years) to occur at 87.0 years. Thus Planet 9 results in a period ratio of 176.5/87.0 = 2.03 for the two components rather than exactly 2.00 and the difference in period from exact harmonic ratio is sufficient to give the slowly varying pattern in Figure 10.

The spectral content of the reconstructed SSN, −6755 to 1885, shown in Figure 11 was obtained after removing a running average over 2400 years from the reconstructed SSN [56]. The spectrum is similar to that obtained with ¹⁴C data [59].

It is clear by comparing the spectrum of SIM, Figure 9(a) and the spectrum of SSN, Figure 11 that the hypothetical transformation of SIM into sunspot emergence, if it exists, acts as a low pass filter, i.e. the transform of cycles of SIM into cycles of sunspot emergence is much stronger at low frequencies than at high frequencies. The periods of reconstructed SSN in Figure 11 along with the apparently corresponding periods of R_B/R_SUN in brackets are: 2160 (2400), 1440 (?), 960 (890), 720 (667), 617 (?), 508 (535), 346 (380), 206 (208), ??? (168), 149 (?), 133 (?), 104 (98), 87 (85), 68 (62), 53 (55), 43 (45), 38 (36), 33 (?), 29 (29.6). Where there is no apparent correspondence a question mark is placed. Note that the three question marks indicate the absence of the ~168 year Jose component from the SSN record. Clearly there is a reasonably close correspondence between the low frequency periodicity of SIM and reconstructed SSN apart from the Jose component that is marked with the dotted reference lines in Figure 11. This issue, which challenges the hypothesis of a connection between SIM and solar activity, is discussed in detail below.

The modulation of a high frequency signal by a lower frequency signal is the basis of communication engineering where the higher frequency signal is called the carrier and the lower frequency signal is called the modulation. The basic types of modulation used in communications are amplitude modulation and phase/frequency modulation. The same concept has been applied to investigating the spectral content of solar activity and climate variables, e.g. amplitude modulation [60] and phase/frequency modulation [61] [62]. In amplitude modulation a fraction of the power at the carrier frequency is shifted into sidebands on either side of the carrier frequency while in phase/frequency modulation most of the power is shifted from the carrier frequency into sidebands of the carrier frequency. The latter occurs, for example, in the modulation of the annual cycle of the interplanetary magnetic field by the ~22 year Hale cycle, where all the power in the annual cycle is shifted into sidebands [61]. So a characteristic of the spectra resulting from phase/frequency modulation is a distinct minimum at the carrier frequency and strong sidebands spaced equally at each side of the carrier frequency. There is a deep minimum at ~170 years in spectrum of reconstructed sunspot number, Figure 11. A possible explanation for the absence of the Jose periodicity at ~170 years from the spectrum is that the Jose periodicity is split into sidebands by phase modulation by lower frequency components. We investigate this possibility in the following.

Modulation of a carrier signal of angular frequency, ω_C, by a lower angular frequency, ω_M, modulating signal can be represented by

y ( t ) = [ A + cos ( ω M t ) ] cos ( ω C t )

y ( t ) = A cos ( ω C t ) + 0.5 [ cos ( ( ω C + ω M ) t ) + cos ( ( ω C − ω M ) t ) ]

y ( t ) = A cos ( ω C t ) + 0.5 [ cos ( ω H t ) + cos ( ω L t ) ] (5)

where ω_H and ω_L are the angular frequencies of the high and low frequency sidebands respectively. When A > 1 we have amplitude modulation, A < 1 partial amplitude and phase modulation, and A = 0, pure phase modulation where on every second half cycle of the modulating signal the phase of the carrier signal is reversed. If f_H and f_L can be found the modulating frequency is,

f M = ( f H − f L ) / 2 (6)

and the carrier frequency is,

f C = ( f H + f L ) / 2 (7)

When A = 0 we have pure phase modulation. As an example, if there are two components in the modulating signal, at periods 1500 years and 600 years, and the carrier period is 168 years, the Jose periodicity, the modulated wave form and the spectrum are as shown in Figure 12(a) and Figure 12(b).

In this case, the carrier frequency is easy to determine as the midpoint frequency between the sidebands in the spectrum. If the carrier period and the carrier phase are known, the low frequency modulating waveform can be recovered by demodulating the modulated signal, the variation in (A) in the Figure 12. Demodulation is obtained by correlating the modulated signal with the carrier signal, in the above simulation the carrier signal was cos(2πt/168 + 0). The result of the demodulation is shown in Figure 12(c) which is indicative of the modulated signal alternating between in-phase and out-of-phase with the Jose cycle. The spectrum of the demodulated signal is shown in Figure 12(d). Note that periods of the original modulating cycles, 1500 year and 600 year, are, by this process, recovered from the modulated waveform. Clearly the demodulation method is not exact and weak spurious components are generated as evident in Figure 12(d). It is also possible, by using slight variations in period and phase of the Jose cycle, and the above simulation methods, to show that the demodulation method is very sensitive to the period and phase of the Jose cycle.

It is clear from the above that it is possible to move back and forth between a modulated signal and the modulating signal. We now apply this method to the reconstructed SSN record [56] with the objective of demonstrating the possibility that the Jose cycle is the “hidden” mediator between the low frequency and mid frequency components of the SSN record. Figure 13 outlines the transformation by which the low frequency components of the SSN variation modulate

the Jose cycle and result in the mid frequency components of the SSN. The very low frequency component of SSN record, period longer than ~2400 years, that Usoskin et al. [22] regarded as likely being of terrestrial origin, was removed by subtracting a 2400 year running average. This resulted in the very complex blue broken curve shown in Figure 13(a). A running average over 30 points, or 300 years, was then used to obtain the low frequency waveform, the full line in Figure 13(a). The spectrum of this low frequency SSN waveform, shown in Figure 13(b), has seven significant low frequency components. Figure 13(c) is the result of modulating the Jose cycle, cos(2πt/168 + π), obtained from Figure 10(a), with the low frequency waveform of Figure 13(a). The resulting spectrum, corresponding to variation in the mid frequency range of the SSN record is shown in Figure 13(d), the most noticeable feature being the distinct minimum in spectral amplitude centred on the Jose period, 168 years. This spectrum, Figure 13(d), is directly comparable, in the same frequency range with Figure 11(b), which shows the entire SSN spectrum. The sidebands in Figure 13(d) are evidently the result of phase modulation of the Jose cycle by the low frequency components present in the variation in Figure 13(a). For example the most central pair of sidebands in 13 D is due to modulation by the 0.0005 year⁻¹, ~2000 year, component in Figure 13(a) and the second set of sidebands from the centre is due to phase modulation by the ~1000 year component in Figure 13(a).

We now demonstrate the demodulation of the SSN record in the mid frequency range via correlation with the Jose cycle to obtain the low frequency components of the reconstructed SSN record [56]. The mid frequency range variation of the SSN record is obtained by using a digital band pass filter to isolate the time variation of SSN in the frequency range from 0.003 years⁻¹ to 0.009 years⁻¹, Figure 14(a). The spectrum of this variation, shown in Figure 14(b), has all the low frequency components of the SSN variation removed by the filter. We now correlate the mid frequency range variation, Figure 14(a), with the Jose cycle, cos(2πt/168 + π), as determined in Figure 10(a). The variation of the resulting correlation coefficient, after smoothing with a 100 year running average, is shown in Figure 14(c). The spectrum of this variation, shown in Figure 14(d), contains low frequency components similar in period to the low frequency components of the SSN record in Figure 13(b).

The time variation in Figure 14(c) and the spectrum in Figure 14(d) are reasonably consistent with the low frequency time variation of Figure 13(a) and the low frequency spectrum of Figure 13(b), indicating that the demodulation of the mid frequency range variation in the SSN record with the Jose cycle is recovering, to a reasonable approximation, the low frequency time variation of the reconstructed SSN record. The effectiveness of the type of transformations illustrated in Figure 13 and Figure 14 support the concept that phase modulation of the Jose cycle in SIM by the low frequency components in SIM results in the mid frequency range variation of the reconstructed SSN. It also explains the occurrence of the deep minimum in the spectrum of the SSN record at ~170 years period that is evident in the spectrum of reconstructed SSN, Figure 11(a) and

Figure 11(b), and in previous observations of the spectral content of cosmogenic isotope series [55] [60] [63]. Note that in a later section, Section 6.4, we find that the Jose periodicity in SIM varies slightly with Planet 9 orbital radius and, if the Planet 9 orbit is eccentric, an exactly constant Jose periodicity may not apply to SIM.

The hypothetical connection between SIM and SSN is that SIM, by some mechanism, as discussed below, induces changes in the magnetically active region of the Sun that vary the rate of emergence of sunspots. The connection between SIM and the reconstructed SSN involves several further connections. Connections between sunspots and solar wind, solar wind and cosmic rays, cosmic rays and the production of ¹⁴C or ¹⁰Be isotopes, the accumulation of the isotopes in tree rings or ice layers, the interpretation of tree rings and ice layers into a time sequences, and the conversion of the resulting cosmogenic time series into a proxy record of sunspot emergence [56]. In view of the noise, non-linearities and uncertainties involved in such a complex process it is unlikely that the analysis, outlined above, based on phase modulation of the Jose cycle, Figure 13 and Figure 14, would yield a very close resemblance between the observed and estimated low and mid frequency components of the reconstructed SSN. Nevertheless the comparison is, on close examination, reasonably convincing and we conclude, tentatively, that in the transformation from SIM to solar SSN, the Jose cycle in SIM is phase modulated in the process and that this phase modulation could be the reason for the absence of Jose periodicity in the spectra of SSN.

Grand solar minima have been a preoccupation of solar and climate researchers since the reporting of the Maunder solar minimum by Eddy [20]. The study of grand solar minima is important in the context of understanding the origin of long term variations in the solar dynamo [15] [18] [22] [64] [65] and in relation to the possible climate impacts of grand solar minima, e.g. [55] [66] [67] [68] [69]. The cosmogenic record has enabled the study of the occurrence of grand solar minima over the last ten millennia with [44] reporting the centre times of grand minima in the reconstructed SSN and [53] reporting the centre times of grand minima in the solar modulation potential. The conclusions of the latter two studies is that, “the occurrence of grand minima and maxima is not a result of long-term cyclic variability but is defined by stochastic/chaotic processes” and “other than a weak quasi-periodicity of 2000-2400 years… No other periodicities are observed in the occurrence rate of grand minima” [44]. The conclusions are counter to the findings in this section that grand solar minima result from the phase modulation of the ~170 year Jose cycle by lower frequency cycles such as the ~1000 year Eddy and ~2400 year Hallstatt cycles. However, Inceoglu et al. [64] have changed their view about grand solar minima to, “it would also imply that these quiescent periods of activity are not the result of a random process but instead their origin is linked to the driving mechanism of magnetic field generation”.

With a view to assessing if long term cyclic variability is present or not in the occurrence of grand minima we here study the central dates of grand solar minima observed by [44] [53]. The objective is to assess if the ~170 year Jose cycle and the harmonic and sub harmonics of the Jose cycle are evident in the occurrence dates. The method is to specify, in the relevant time interval, a value unity for the centre years of grand solar minima and a value of zero for all other years. To enable the use of FFT analysis the central dates of grand solar minima are assigned to the nearest fifth year in the sequence of years between −9200 and 1810 and given the value unity. The sequence, after smoothing with a ten year running average, and representing the occurrence of grand minima as specified by [44] or [53], is shown in Figure 15(a) for the [44] data and Figure 16(a) for

the [53] data. A fast Fourier transform of the sequence is then made to identify periodicities in the sequences, Figure 15(b) and Figure 16(b). This method is similar to that applied to uncover the spectral content of ground level enhancement events [70]. The strongest peak in the spectrum of the data recorded in [44] is at 352.9 years. We associate this period with 2TJ where TJ is the Jose periodicity, 176.45 years. As expected from the discussion about phase modulation, a prominent peak at period TJ is not evident in the spectrum and the period TJ is associated with a deep broad minimum in the spectrum. Four sub-harmonics, at periods 2.5TJ, 3TJ, 4TJ, and 6TJ coincide with prominent peaks and are indicated with reference lines. The first harmonic of the Jose periodicity at 0.5TJ = 88.22 years is also indicated. The long term periodicity at ~2400 years that represents the clustering of solar grand minima occurrences is also marked. Note that the ~2400 year cycle of clusters of grand solar minima in Figure 15(a) corresponds to the clusters occurring within the negative phase of the ~2400 year Hallstatt cycle in SIM in Figure 10(a).

The analysis is repeated for the reconstructions of solar modulation potential from the ¹⁰Be and ¹⁴C cosmogenic series [53]. This sequence of grand solar minima occurrences is shown in Figure 16(a) and the resulting spectral content of the occurrences is shown in Figure 16(b). The periodicities marked by reference lines are essentially the same as in the spectrum of Figure 15(b) derived from the [44] data. Thus three different estimates of the occurrence of grand minima give essentially the same periodicity of occurrence and that periodicity is dominated by a Jose periodicity of ~177 years and sub harmonics of this periodicity.

The ~177 year Jose periodicity in the occurrence of solar grand minima, while clearly associated with the low frequency components in the spectrum is “hidden” in the sense that the amplitude of the component is relatively weak compared with the amplitude of the sidebands and with the amplitude of lower frequency components. This is a result, as outlined in Section 6.2, of phase modulation of the Jose component by lower frequency components. However, the phase modulated component is not completely “hidden” if the number of phase reversed cycles or events differs significantly from the number of regular cycles or events. In the context of grand solar minima occurrences, if all grand minima occurred at n × 176 year intervals in the time sequence then there would be a strong peak at 176 years in the spectrum of occurrences. If, however, about half of the grand minima occurred at (n + ½) × 176 year intervals in the time sequence the 176 year periodicity in the spectrum would be “hidden”. In the spectrum of Figure 16(b) a peak at ~177 years is evident but relatively weak indicative of an imbalance between in phase and out-of-phase occurrences of grand solar minima. There are several strong peaks in the spectrum that correspond to phase modulation of the Jose cycle that have not been marked by reference lines. For example, using Equations (6) and (7), the peaks at 303 years, (0.0033 years⁻¹), and 256 years, (0.0039 years⁻¹), in Figure 16(b) are sideband peaks due to phase modulation of the Jose cycle by the 442 year period and 531 year period low frequency components respectively. Similarly the strong, broad peaks at ~0.005 year⁻¹ (200 years) and ~0.007 years⁻¹ ( 143 years) are sideband peaks due to phase modulation of the ~177 year Jose cycle by the ~1000 year Eddy cycle.

Beer et al. [27] suggested that solar grand minima “recur with the characteristic de Vries period of approximately 208 years” and supported this idea by band pass filtering the cosmogenic ¹⁴C record in the period range between 180 and 230 years, further stating that the maxima in the filtered cosmogenic record “actually correspond to grand minima in solar activity”. The evidence of Figure 16(b) suggests that the recurrence time of solar grand minima obtained from the ¹⁴C record is close to 177 years. If the recurrence time was 208 years a peak at 0.0048 years⁻¹ would be evident in the spectrums in Figure 15(b) and Figure 16(b).

The above analysis is presented as a possible reason why the Jose periodicity, a dominant periodicity in the spectrum of SIM, is absent from the spectrum of reconstructed SSN whereas other periodicities in SIM, e.g. the ~2400 year Hallstatt, ~1000 year Eddy, and the ~88 year Gleissberg periodicities, are clearly present in the spectrum of reconstructed SSN, c.f. Figure 11(a) and Figure 11(b).

Another conundrum in this attempt to relate SIM to SSN is the discrepancy between the ~177 year Jose periodicity obtained by spectral analysis of the solar activity record, Figure 11(a) and Figure 16(b), and the ~170 year Jose periodicity obtained from SIM, Figure 9(a) and Figure 10(a). Sharp [71] and McCracken et al. [32], using very long records of SIM, estimated the Jose period to be ~171 years as opposed to the estimate by Jose (1965), over the short interval 1653 to 2060, of a Jose period of ~about 178 years. The estimate obtained in this paper using the approximation of circular planet orbits is a period ~169 years for SIM including Planet 9 and a period ~172 years for SIM for the eight planet system, the latter value consistent with the estimate by Sharp [71] and McCracken et al. [32]. The discrepancy between ~177 years for SSN and ~170 years for SIM is large enough to be a significant challenge to the planetary hypothesis.

The previous section found the “hidden” Jose periodicity in SSN is ~176.5 years, Figure 15 and Figure 16, whereas the calculated Jose periodicity in SIM was ~168.5 years, Figure 9(a). This discrepancy may be resolved by noting that the Jose periodicity in SIM varies with the semi-major axis, a₉, of Planet 9, Figure 17.

In fact all of the SIM periodicities that depend on the semi-major axis of Planet 9 vary. The Gleissberg ~88 year periodicity in SIM also decreases with a₉, Figure 17. However, in contrast, the Hallstaat periodicity in SIM increases strongly with a₉, Figure 9(a). Thus, a possible resolution of the difference between the SSN and SIM estimates of Jose periodicity is that the orbital radius of Planet 9 used in the calculation of SIM in this paper, 380 AU, the best estimate from [1], is actually shorter, about 250 AU.

Another possibility is that the Planet 9 orbit is eccentric of the order e ~ 0.3, and the occurrence of grand solar minima cluster around the times of perihelion when the orbital radius approaches a lower value of 250 AU. In this case most of the solar grand minima in SSN would be expected to occur, in clusters, at a time spacing of about 176 years, as observed, corresponding to the Jose periodicity in SIM when a₉ ~ 250 AU, also ~period 176 years.