Prediction of future fifteen solar cycles
K. M. Hiremath
Indian Institute of astrophysics, Bangalore-560034, India
hiremath@iiap.res.in
ABSTRACT
In the previous study (Hiremath 2006a), the solar cycle is modeled as a forced
and damped harmonic oscillator and from all the 22 cycles (1755-1996), long-
term amplitudes, frequencies, phases and decay factor are obtained. Using these
physical parameters of the previous 22 solar cycles and by an autoregressive
model, we predict the amplitude and period of the future fifteen solar cycles.
Predicted amplitude of the present solar cycle (23) matches very well with the
observations. The period of the present cycle is found to be 11.73 years. With
these encouraging results, we also predict the profiles of future 15 solar cycles.
Important predictions are : (i) the period and amplitude of the cycle 24 are 9.34
years and 110 (±11), (ii) the period and amplitude of the cycle 25 are 12.49 years
and 110 (± 11), (iii) during the cycles 26 (2030-2042 AD), 27 (2042-2054 AD),
34 (2118-2127 AD), 37 (2152-2163 AD) and 38 (2163-2176 AD), the sun might
experience a very high sunspot activity, (iv) the sun might also experience a very
low ( around 60) sunspot activity during cycle 31 (2089-2100 AD) and, (v) length
of the solar cycles vary from 8.65 yrs for the cycle 33 to maximum of 13.07 yrs
for the cycle 35.
Subject headings: sunspots – solar cycle – prediction
1. Introduction
Owing to proximity, the sun influences the earth’s climate and environment. Over-
whelming evidence is building up that the solar cycle and related activity phenomena are
correlated with the earth’s global climate and temperature, the sea surface temperatures of
the three (Atlantic, Pacific and Indian) main ocean basins, the earth’s albedo, the galactic
cosmic ray flux that in turn is correlated with the earth’s cloud cover and, Indian monsoon
rainfall (Hiremath and Mandi 2004 and references there in; Georgieva et. al. 2005; Hire-
math 2006b). The transient parts of the solar activity such as the flares and the coronal
http://arxiv.org/abs/0704.1346v1
– 2 –
mass ejections that are directed towards the earth create havoc in the earth’s atmosphere by
disrupting the global communication, reducing life time of the earth bound satellites and,
keep in dark places of the earth that are at higher latitudes by breaking the electric power
grids. Owing to sun’s immense influence of space weather effects on the earth’s environment
and climate, it is necessary to predict and know in advance different physical parameters
such as amplitude and period of the future solar cycles.
There are many predictions in the literature (Ohl 1966; Feynman 1982; Feynman and
Gu 1986; Kane 1999; Hathaway, Wilson and Reichmann 1999; Badalyan, Obrido and Sykora
2001; Duhau 2003 Sello 2003; Maris, Poepscu and Besliu 2003; Euler and Smith 2004; Maris,
Poepscu and Besliu 2004; Kaftan 2004; Echer et. al. 2004; Gholipour et. al., 2005; Schatten
2005; Li, Gao and Su 2005; Svaalgaard, Cliver and Kamide 2005; Chopra and Dabas 2006;
Dikpati, Toma and Gilman 2006; Du 2006; Hathaway and Wilson 2006; Clilverd et. al., 2006;
Tritakis and Vasilis 2006; Lantos 2006; Lundstedt 2006; Wang and Sheeley 2006; Choudhuri,
Chatterjee and Jiang 2007; Javaraiah 2007) on the previous and future 24th solar cycles and
beyond. Most of these studies mainly concentrate on prediction of the amplitude (maximum
sunspot number during a cycle). However, prediction of period (length) of a solar cycle is
also very important parameter and the present study fills that gap.
Recently we modeled the solar activity cycle as a forced and damped harmonic oscillator
that consists of both the sinusoidal and transient parts (eqn 1 of Hiremath 2006a). From
the 22 cycles (1755-1996) sunspot data, the physical parameters (amplitudes, frequencies,
phases and decay factors) of such a harmonic oscillator are determined. The constancy of the
amplitudes and the frequencies of the sinusoidal part and a very small decay factor from the
transient part suggests that the solar activity cycle mainly consists of persistent oscillatory
part that might be compatible with long-period (∼ 22 yrs) Alfven oscillations. In the present
study, with an autoregressive model and by using the physical parameters of 22 cycles, we
predict the amplitudes and periods of future 16 solar cycles. Thus prediction from this study
can be considered as a physical and precursor method.
A Pth order autoregressive model relates a forecasted value xt of the time series X =
[x0, x1, x2, ..., xt−1], as a linear combination of P past values xt = φ1xt−1 + φ2xt−2 + ...... +
φpxt−p +Wt , where the coefficients φ1, φ2, ..., φp are calculated such that they minimize the
uncorrelated random error terms, Wt. The routine is available in IDL software. Important
condition for using an autoregressive model is that the series must be stationary such that
it’s mean and standard deviation do not vary much with time. Hence, one can not apply
autoregressive model directly to the observed sunspot series as it consists of near sinusoidal
trends whose amplitudes and the standard deviations entirely different for different solar
cycles. On the other hand, the derived physical parameters of the forced and damped
– 3 –
harmonic oscillator (Hiremath 2006a) for all the 22 solar cycles are stationary and, hence in
the following, we use an autoregressive model to predict the future 15 solar cycles.
The solution of the forced and damped harmonic oscillator (see the equation 1 of Hir-
math 2006a) of the solar cycle consists of two parts : (i) the sinusoidal part that determines
the amplitude and period of the solar cycle and, (ii) the transient part that dictates decay
of the solar cycle from the maximum year and also determines bimodal structure of the
sunspot cycle around the maximum years for some cycles. In the present study, we use
physical parameters of the sinusoidal part only to predict amplitude and period of future
cycles.
2. Results and conclusion
Using past 22 cycles’ physical parameters, we construct the next (23rd) solar cycle
and presented in Fig 1. Except decaying part of the solar cycle, one can notice that the
predicted curve exactly matches with the observed curve. With this encouragement and from
an autoregressive model, the physical parameters of future 16 solar cycles are computed and
reconstructed solar cycles are presented in Fig 2. For the coming cycles 24-38, the results are
summarized in Table 1. In Table 1, the first column represents the cycle number, the second
column represents the year from minimum-minimum, the third column represents the period
(length) of the solar cycle and, the last column represents the maximum sunspot number
during a cycle. It is interesting to note that the amplitude of the cycle 24 is low compared
to the amplitude of cycle 23 and is almost similar to average value computed from all of
the predicted models (http://members.chello.be/j.janssens/SC24.html). Other interesting
predictions are : (i) during the cycles 26, 27, 34. 37 and 38, the sun will experiences a very
high solar activity, (ii) during cycle 31 (2087-2099 AD) the sun will experiences a very low
sunspot activity and, (iii) length of the solar cycles vary from 8.65 yrs for the cycle 33 to
maximum of 13.07 yrs for cycle 25.
To conclude, the solar cycle is modeled as a forced and damped harmonic oscillator.
From the previous 22 cycles sunspot data, the physical parameters such as the amplitudes,
the frequencies and phases of such a harmonic oscillator are determined. The sinusoidal
part of the forced and damped harmonic oscillator of previous solar cycles is considered
for the prediction of future 16 cycles. With an autoregressive model and using previous 22
cycles parameters, coming 16 solar cycles are reconstructed from the predicted parameters.
Important results of this prediction are : the amplitude of coming solar cycle 24 will be
smaller than the present cycle 23 and around 2087-2099 AD, the sun will experiences a very
low sunspot activity.
http://members.chello.be/j.janssens/SC24.html
– 4 –
The author is thankful to Dr. Luc Dame and Dr. Javaraiah for the useful discussions.
REFERENCES
Badalyan, O. G., Obrido, V. N & Sykora, J. 2001, solar physics, 199, 421
Chopra, P & Dabas, R. S. 2006, in the Cospar Proceedings, Beijing
Choudhuri, A. R., Chatterjee, P & Jiang, J. 2007, astro-ph/0701527
Clilverd, M. A., Clarke, E., Ulich, T., Rishberth, H & Jarvis, M. J., 2006. Space Weather,
vol 4, S09005
Dikpati, M., Toma, G. D & Gilman, P. 2006, Geophys. Res. Lett., 33, L05102
Du, Z. L. 2006, A&A, 457, 309
Duhau, S. 2003, Sol. Phys., 213, 203
Echer, E., Rigozo, N. R., Nordemann, D. J. R & Vieira, I. E. A. 2004, Annales Geophysicae,
22, 2239
Feynman, J. 1982, JGR, 87, 6153
Feynman, J & Gu, X. Y. 1986, Rev. Geophys., 24, 650
Georgieva, K., Kiro, B., Javaraiah, J & Krasteva, R. 2005, Planet. Space. Sci., 53, 197
Gholipour, A., Lucas, C., Arabi, B. N & Shafiee, M. 2005, Solar. Terr. Phys, 67, 595
Javaraiah, J. 2007, MNRAS LETTERS, DOI: 10.1111/j.1745-3933.2007.00298.x
Hathaway, D., Wilson, R. M & Reichmann, J. 1999, JGR, 104, 375
Hathaway, D & Wilson, R. M. 2006, Geophys. res. lett., 33, L18101
Hiremath, K, M & Mandi, P. I. 2004, New Astronomy, 9, 651
Hiremath, K. M. 2006a, A&A, 452, 591
Hiremath, K. M. 2006b, ILWS Workshop, Goa, India, Edts : Gopalswamy, N and Bhat-
tacharya, A., p. 178
Kaftan, V. 2004, in the proceedings of IAU symp 223, p. 111
http://arxiv.org/abs/astro-ph/0701527
– 5 –
Kane, R. P. 1999, Sol. Phys., 189, 217
antos, S. 2006, Sol. Phys., 236, 399
undstedt, H. 2006, Advance. Space. Res., 38, 862
Mari, S. G., Popescu, M. D., & Besliu, D. 2003, Romanian. Astron. Journ., 13, 139
Mari, S. G., Popescu, M. D., & Besliu, D. 2004, in the proceedings of IAU Symp 233
Li, K.J., Gao, P. X & Su, T.w. 2005, Chin. J. Astron. Astrophys., vol 5, No 5, 539
Ohl, A. I. 1966, Sol. Dannye, No. 12, 84
Sello, S. 2003, A & A, 410, 691
Schatten, K. 2005, Geophys. Res. Let, 32, L21106
Svalgaard, L., Cliver, E. W & Kamide, Y. 2005, Geophysical. Res. Let, 32, L001104
Vasilis, T., Helen, M & George, G. 2006, in the proceedings of AIP conference, 848, 154
ang, Y. M & Sheeley, Jr, N. R. 2006, Nature Phys, vol 2, Issue 6, 367
This preprint was prepared with the AAS LATEX macros v5.2.
– 6 –
Fig. 1.— Prediction of the future solar cycles. (a) The left plot illustrates the predicted (red
continuous) curve over plotted on the observed (blue curve) sunspot cycle 23. The dashed
red curves represent uncertainty in the Prediction. (b) The right plot illustrates predicted
future 15 solar cycles. The red numbers over different solar cycle maximum are the cycle
numbers.
– 7 –
Table 1. Predicted sunspot cycles
Cycle Year Period Maximum
Number Min-Min (Years) Number
23 1996.00-2007.73 11.73 136±14
24 2007.73-2017.07 9.34 110±11
25 2017.07-2029.56 12.49 110±11
26 2029.56-2041.50 11.94 157±16
27 2041.50-2053.51 12.00 180±18
28 2053.51-2064.30 10.80 140±14
29 2064.30-2075.01 10.71 149±15
30 2075.01-2086.79 11.78 118±12
31 2086.79-2097.95 11.16 63±6
32 2097.95-2108.84 10.89 108±11
33 2108.84-2117.49 8.65 128±13
34 2117.49-2126.92 9.43 170±17
35 2126.92-2139.99 13.07 139±14
36 2139.99-2151.74 11.75 159±16
37 2151.74-2163.19 11.45 187±19
38 2163.19-2175.48 12.29 187±19
	Introduction
	Results and conclusion

  In the previous study (Hiremath 2006a), the solar cycle is modeled as a
forced and damped harmonic oscillator and from all the 22 cycles (1755-1996),
long-term amplitudes, frequencies, phases and decay factor are obtained. Using
these physical parameters of the previous 22 solar cycles and by an {\em
autoregressive model}, we predict the amplitude and period of the future
fifteen solar cycles. Predicted amplitude of the present solar cycle (23)
matches very well with the observations. The period of the present cycle is
found to be 11.73 years. With these encouraging results, we also predict the
profiles of future 15 solar cycles. Important predictions are : (i) the period
and amplitude of the cycle 24 are 9.34 years and 110 ($\pm 11$), (ii) the
period and amplitude of the cycle 25 are 12.49 years and 110 ($\pm$ 11), (iii)
during the cycles 26 (2030-2042 AD), 27 (2042-2054 AD), 34 (2118-2127 AD), 37
(2152-2163 AD) and 38 (2163-2176 AD), the sun might experience a very high
sunspot activity, (iv) the sun might also experience a very low (around 60)
sunspot activity during cycle 31 (2089-2100 AD) and, (v) length of the solar
cycles vary from 8.65 yrs for the cycle 33 to maximum of 13.07 yrs for the
cycle 35.

Introduction
Owing to proximity, the sun influences the earth’s climate and environment. Over-
whelming evidence is building up that the solar cycle and related activity phenomena are
correlated with the earth’s global climate and temperature, the sea surface temperatures of
the three (Atlantic, Pacific and Indian) main ocean basins, the earth’s albedo, the galactic
cosmic ray flux that in turn is correlated with the earth’s cloud cover and, Indian monsoon
rainfall (Hiremath and Mandi 2004 and references there in; Georgieva et. al. 2005; Hire-
math 2006b). The transient parts of the solar activity such as the flares and the coronal
http://arxiv.org/abs/0704.1346v1
– 2 –
mass ejections that are directed towards the earth create havoc in the earth’s atmosphere by
disrupting the global communication, reducing life time of the earth bound satellites and,
keep in dark places of the earth that are at higher latitudes by breaking the electric power
grids. Owing to sun’s immense influence of space weather effects on the earth’s environment
and climate, it is necessary to predict and know in advance different physical parameters
such as amplitude and period of the future solar cycles.
There are many predictions in the literature (Ohl 1966; Feynman 1982; Feynman and
Gu 1986; Kane 1999; Hathaway, Wilson and Reichmann 1999; Badalyan, Obrido and Sykora
2001; Duhau 2003 Sello 2003; Maris, Poepscu and Besliu 2003; Euler and Smith 2004; Maris,
Poepscu and Besliu 2004; Kaftan 2004; Echer et. al. 2004; Gholipour et. al., 2005; Schatten
2005; Li, Gao and Su 2005; Svaalgaard, Cliver and Kamide 2005; Chopra and Dabas 2006;
Dikpati, Toma and Gilman 2006; Du 2006; Hathaway and Wilson 2006; Clilverd et. al., 2006;
Tritakis and Vasilis 2006; Lantos 2006; Lundstedt 2006; Wang and Sheeley 2006; Choudhuri,
Chatterjee and Jiang 2007; Javaraiah 2007) on the previous and future 24th solar cycles and
beyond. Most of these studies mainly concentrate on prediction of the amplitude (maximum
sunspot number during a cycle). However, prediction of period (length) of a solar cycle is
also very important parameter and the present study fills that gap.
Recently we modeled the solar activity cycle as a forced and damped harmonic oscillator
that consists of both the sinusoidal and transient parts (eqn 1 of Hiremath 2006a). From
the 22 cycles (1755-1996) sunspot data, the physical parameters (amplitudes, frequencies,
phases and decay factors) of such a harmonic oscillator are determined. The constancy of the
amplitudes and the frequencies of the sinusoidal part and a very small decay factor from the
transient part suggests that the solar activity cycle mainly consists of persistent oscillatory
part that might be compatible with long-period (∼ 22 yrs) Alfven oscillations. In the present
study, with an autoregressive model and by using the physical parameters of 22 cycles, we
predict the amplitudes and periods of future 16 solar cycles. Thus prediction from this study
can be considered as a physical and precursor method.
A Pth order autoregressive model relates a forecasted value xt of the time series X =
[x0, x1, x2, ..., xt−1], as a linear combination of P past values xt = φ1xt−1 + φ2xt−2 + ...... +
φpxt−p +Wt , where the coefficients φ1, φ2, ..., φp are calculated such that they minimize the
uncorrelated random error terms, Wt. The routine is available in IDL software. Important
condition for using an autoregressive model is that the series must be stationary such that
it’s mean and standard deviation do not vary much with time. Hence, one can not apply
autoregressive model directly to the observed sunspot series as it consists of near sinusoidal
trends whose amplitudes and the standard deviations entirely different for different solar
cycles. On the other hand, the derived physical parameters of the forced and damped
– 3 –
harmonic oscillator (Hiremath 2006a) for all the 22 solar cycles are stationary and, hence in
the following, we use an autoregressive model to predict the future 15 solar cycles.
The solution of the forced and damped harmonic oscillator (see the equation 1 of Hir-
math 2006a) of the solar cycle consists of two parts : (i) the sinusoidal part that determines
the amplitude and period of the solar cycle and, (ii) the transient part that dictates decay
of the solar cycle from the maximum year and also determines bimodal structure of the
sunspot cycle around the maximum years for some cycles. In the present study, we use
physical parameters of the sinusoidal part only to predict amplitude and period of future
cycles.
2. Results and conclusion
Using past 22 cycles’ physical parameters, we construct the next (23rd) solar cycle
and presented in Fig 1. Except decaying part of the solar cycle, one can notice that the
predicted curve exactly matches with the observed curve. With this encouragement and from
an autoregressive model, the physical parameters of future 16 solar cycles are computed and
reconstructed solar cycles are presented in Fig 2. For the coming cycles 24-38, the results are
summarized in Table 1. In Table 1, the first column represents the cycle number, the second
column represents the year from minimum-minimum, the third column represents the period
(length) of the solar cycle and, the last column represents the maximum sunspot number
during a cycle. It is interesting to note that the amplitude of the cycle 24 is low compared
to the amplitude of cycle 23 and is almost similar to average value computed from all of
the predicted models (http://members.chello.be/j.janssens/SC24.html). Other interesting
predictions are : (i) during the cycles 26, 27, 34. 37 and 38, the sun will experiences a very
high solar activity, (ii) during cycle 31 (2087-2099 AD) the sun will experiences a very low
sunspot activity and, (iii) length of the solar cycles vary from 8.65 yrs for the cycle 33 to
maximum of 13.07 yrs for cycle 25.
To conclude, the solar cycle is modeled as a forced and damped harmonic oscillator.
From the previous 22 cycles sunspot data, the physical parameters such as the amplitudes,
the frequencies and phases of such a harmonic oscillator are determined. The sinusoidal
part of the forced and damped harmonic oscillator of previous solar cycles is considered
for the prediction of future 16 cycles. With an autoregressive model and using previous 22
cycles parameters, coming 16 solar cycles are reconstructed from the predicted parameters.
Important results of this prediction are : the amplitude of coming solar cycle 24 will be
smaller than the present cycle 23 and around 2087-2099 AD, the sun will experiences a very
low sunspot activity.
http://members.chello.be/j.janssens/SC24.html
– 4 –
The author is thankful to Dr. Luc Dame and Dr. Javaraiah for the useful discussions.
REFERENCES
Badalyan, O. G., Obrido, V. N & Sykora, J. 2001, solar physics, 199, 421
Chopra, P & Dabas, R. S. 2006, in the Cospar Proceedings, Beijing
Choudhuri, A. R., Chatterjee, P & Jiang, J. 2007, astro-ph/0701527
Clilverd, M. A., Clarke, E., Ulich, T., Rishberth, H & Jarvis, M. J., 2006. Space Weather,
vol 4, S09005
Dikpati, M., Toma, G. D & Gilman, P. 2006, Geophys. Res. Lett., 33, L05102
Du, Z. L. 2006, A&A, 457, 309
Duhau, S. 2003, Sol. Phys., 213, 203
Echer, E., Rigozo, N. R., Nordemann, D. J. R & Vieira, I. E. A. 2004, Annales Geophysicae,
22, 2239
Feynman, J. 1982, JGR, 87, 6153
Feynman, J & Gu, X. Y. 1986, Rev. Geophys., 24, 650
Georgieva, K., Kiro, B., Javaraiah, J & Krasteva, R. 2005, Planet. Space. Sci., 53, 197
Gholipour, A., Lucas, C., Arabi, B. N & Shafiee, M. 2005, Solar. Terr. Phys, 67, 595
Javaraiah, J. 2007, MNRAS LETTERS, DOI: 10.1111/j.1745-3933.2007.00298.x
Hathaway, D., Wilson, R. M & Reichmann, J. 1999, JGR, 104, 375
Hathaway, D & Wilson, R. M. 2006, Geophys. res. lett., 33, L18101
Hiremath, K, M & Mandi, P. I. 2004, New Astronomy, 9, 651
Hiremath, K. M. 2006a, A&A, 452, 591
Hiremath, K. M. 2006b, ILWS Workshop, Goa, India, Edts : Gopalswamy, N and Bhat-
tacharya, A., p. 178
Kaftan, V. 2004, in the proceedings of IAU symp 223, p. 111
http://arxiv.org/abs/astro-ph/0701527
– 5 –
Kane, R. P. 1999, Sol. Phys., 189, 217
antos, S. 2006, Sol. Phys., 236, 399
undstedt, H. 2006, Advance. Space. Res., 38, 862
Mari, S. G., Popescu, M. D., & Besliu, D. 2003, Romanian. Astron. Journ., 13, 139
Mari, S. G., Popescu, M. D., & Besliu, D. 2004, in the proceedings of IAU Symp 233
Li, K.J., Gao, P. X & Su, T.w. 2005, Chin. J. Astron. Astrophys., vol 5, No 5, 539
Ohl, A. I. 1966, Sol. Dannye, No. 12, 84
Sello, S. 2003, A & A, 410, 691
Schatten, K. 2005, Geophys. Res. Let, 32, L21106
Svalgaard, L., Cliver, E. W & Kamide, Y. 2005, Geophysical. Res. Let, 32, L001104
Vasilis, T., Helen, M & George, G. 2006, in the proceedings of AIP conference, 848, 154
ang, Y. M & Sheeley, Jr, N. R. 2006, Nature Phys, vol 2, Issue 6, 367
This preprint was prepared with the AAS LATEX macros v5.2.
– 6 –
Fig. 1.— Prediction of the future solar cycles. (a) The left plot illustrates the predicted (red
continuous) curve over plotted on the observed (blue curve) sunspot cycle 23. The dashed
red curves represent uncertainty in the Prediction. (b) The right plot illustrates predicted
future 15 solar cycles. The red numbers over different solar cycle maximum are the cycle
numbers.
– 7 –
Table 1. Predicted sunspot cycles
Cycle Year Period Maximum
Number Min-Min (Years) Number
23 1996.00-2007.73 11.73 136±14
24 2007.73-2017.07 9.34 110±11
25 2017.07-2029.56 12.49 110±11
26 2029.56-2041.50 11.94 157±16
27 2041.50-2053.51 12.00 180±18
28 2053.51-2064.30 10.80 140±14
29 2064.30-2075.01 10.71 149±15
30 2075.01-2086.79 11.78 118±12
31 2086.79-2097.95 11.16 63±6
32 2097.95-2108.84 10.89 108±11
33 2108.84-2117.49 8.65 128±13
34 2117.49-2126.92 9.43 170±17
35 2126.92-2139.99 13.07 139±14
36 2139.99-2151.74 11.75 159±16
37 2151.74-2163.19 11.45 187±19
38 2163.19-2175.48 12.29 187±19
	Introduction
	Results and conclusion
