2014–2015 MULTIPLE OUTBURSTS OF 15P/FINLAY

Figure 1 shows time-series false-color composite images at the R_C-band. We chose these images because the R_C-band is the most sensitive to cometary dust among the available filters. In fact, we examined the contribution of the spherical gas component by using the same technique as that described in Section 3.1 of Ishiguro et al. (2014). We found that gas intensity took up only 10 ± 2% (within an aperture at ρ = 10⁴ km from the nucleus) of R_C-band total intensity on UT 2014 December 26, which supports the weak gas flux contribution shown in Figure 1. Some images taken in close time intervals are similar and are thus not shown separately. In the first image captured on UT 2014 December 23 (Figure 1(a)), an envelope structure extends approximately toward the anti-solar direction. It is likely that the envelope is related to the first outburst around UT 2014 December 16. The surface brightness of the envelope reduced quickly and became undetectable after around UT 2014 December 29 (Figure 1(c)). After that time, a near nuclear dust coma and dust tail remained. The coma and tail are attributed to steady activity of the comet because both the shape and magnitude were almost constant over several months. A dramatic change was observed in images after UT 2015 January 16. The inner coma brightened on UT 2015 January 16 (Figure 1(j)), and dust ejecta appearing soon afterward were stretched toward the anti-solar direction (Figures 1(k)–(n)). The appearance of the dust cloud on UT 2015 January 23 (Figure 1(n)) is similar to the image taken on UT 2014 December 23 (Figure 1(a)), in which the comet was enclosed by a widely expanded envelope. The envelope dimmed quickly, leaving behind a near-nuclear dust cloud similar to that from the pre-second outburst. In summary, we observed at least two outburst ejecta superposed on the continuous activity of the comet during its perihelion passage.

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To clarify the time variation of the activity in a more quantitative manner, we conducted aperture photometry of the dust particles in the inner coma. We set a constant physical aperture distance from the nucleus at $\rho ={10}^{4}$ km and integrated the signal within ρ by using the IRAF/APPHOT package. The aperture distances correspond to 74–99 on the sky plane, which is large enough to enclose the seeing disk sizes of these data (typically 2''–3'') but small enough to detect daily changes in dust production (an ejection speed of ∼100 m s⁻¹ was assumed). In general, the measured magnitudes are determined not only by the time-variable activity of comets but also by the observing geometry (i.e., distances and viewing angles). To correct the latter effect, we converted the observed magnitudes into absolute values, which are magnitudes at a unit heliocentric distance r_h = 1 au, observer's distance Δ = 1 au, and solar phase angle α = 0°. These values were determined by

$$\begin{eqnarray}&&{m}_{{\rm{R}}}(1,1,0)={m}_{{\rm{R}}}-5\,\,{\mathrm{log}}_{10}({r}_{{\rm{h}}}{\rm{\Delta }})-2.5\,{\mathrm{log}}_{10}\,{\rm{\Phi }}(\alpha ),\end{eqnarray} \tag{ 5 }$$

where m_R and ${m}_{{\rm{R}}}(1,1,0)$ denote the observed and absolute magnitudes in the R_C-band. The third term on the right-hand side is given to correct phase darkening, which is given by

$$\begin{eqnarray}&&2.5\,{\mathrm{log}}_{10}\,{\rm{\Phi }}(\alpha )=b\alpha ,\end{eqnarray} \tag{ 6 }$$

where the constant b was assumed to be a generally quoted value for cometary dust particles, 0.035 mag deg⁻¹ (Lamy et al. 2004, p. 223). Over our observation period, the observed magnitudes were subtracted by 2.2–3.3 mag by Equations (5)–(6) to convert the absolute magnitudes.

Figure 2 shows the absolute magnitudes with respect to (a) the observed time and (b) the true anomaly f; by definition, the perihelion and aphelion occur at f = 0° and 180°, respectively. The magnitude was almost constant or slightly decreased by 0.03 mag day⁻¹ until the day of the second outburst. A minor eruption appears to have occurred on UT 2015 January 7 with a true anomaly of f = 151, brightening the comet by 0.46 ± 0.14 mag. A careful review of the above time-sequence images revealed a sharp tail on UT 2015 January 7–11, as shown in Figures 1(f)–(h). Because it extended toward the position angle of 1584 ± 10, which matches to anti-solar direction on that day (1589), we considered that a sudden ejection of fresh, small grains or ionized particles might have been stretched in that direction soon after the minor eruption on UT 2015 January 7.

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A more outstanding brightening was observed on UT 2015 January 16 (f = 265). The near-nuclear absolute magnitude was m_R(1, 1, 0) = 6.37 ± 0.10. After the brightening, the magnitudes remained smaller than the pre-outburst magnitudes over three days and returned to a normal magnitude within about one week. Although the image on UT 2015 January 23 (Figure 1(n)) showed the widespread envelope, it is likely that a large part of the dust grains expanded beyond the aperture size of our photometry (i.e., $\rho \gt {10}^{4}$ km) and became obscured by a steady stream of ejecta from the nucleus. To eliminate the magnitude excess on UT 2015 January 23, the outburst ejecta should have an ejection speed able to reach the aperture (i.e., $\rho \gt {10}^{4}$ km). We determined that a large fraction of outburst ejecta should have an effective speed of ≳15 m s⁻¹ to reach the aperture radius within one week.

Polarimetric observation was conducted to find the optical similarities and differences between normal cometary dust and the outburst ejecta. The polarimetric data were acquired about 10 days after the first outburst by using NO 1.6 m and HHO 1.5 m telescopes. R_C-band filters were employed for our observations because this band is less contaminated by gaseous components and is sensitive to the solar-like dust spectrum, as mentioned above (Section 3.2). We integrated the signals of the observed data within apertures of ρ = 8000 km and 12,000 km, and derived the polarization degree (P_r) and the position angle of the polarization vector (${\theta }_{{\rm{P}}}$). We obtained P_r = 6.5 ± 0.3% on UT 2014 December 26 and P_r = 7.0 ± 0.5% on UT 2014 December 27. The position angle was aligned to the normal vector of the scattering plane, that is, θ_P = 1° ± 4°. No significant difference was noted within the errors in P_r between measurements with the large and small apertures (i.e., ρ = 8000 km and 12,000 km). Figure 3 compares the polarimetric results with those of other comets. The phase angle dependence of the polarization degree is known to be classified into two groups: dust-rich comets with high polarization degrees and gas-rich comets with low polarization degrees (Levasseur-Regourd et al. 1996). Comet C/1995 O1 (Hale–Bopp) did not match these two categories and showed a very high polarization degree (Hadamcik et al. 1997). The polarization degree of 15P fell into common values of comets between gas-rich and dust-rich groups. As described above, the surface brightness of the outburst ejecta faded out within ∼seven days from the near-nuclear region in the case of the second outburst, which is likely similar to the case of the first outburst, and was indistinguishable from the background dust particles produced by continuous activity. Therefore, the similarity in polarization degree does not mean that the optical properties of outburst ejecta are the same as those in normal comets. As a result of the data analysis, we learned that earlier follow-up observations within ∼three days are required to determine the characteristics of such fresh particles during outbursts, such as fluffy or compact optical properties. This topic should be covered in future observation planning.

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In this section, we provide an approximate but straightforward estimate of outburst ejecta mass to compare with previous outbursts, and subsequently present a more sophisticated estimate by conducting a dynamical analysis of dust particles in Section 3.6. We measured the total R_C-band magnitudes with apertures large enough to enclose the entire visible dust cloud for obtaining the total mass of the outburst materials. We set aperture radii of ρ = 45 (UT 2014 December 23) and 1' (UT 2015 January 16), and obtained m_R = 9.44 ± 0.10 (UT 2014 December 23) and m_R = 7.95 ± 0.10 (UT 2015 January 16). The magnitude was m_R = 11.12 ± 0.12 on UT 2015 January 11–13, which was obtained just prior to the second outburst and well after the first outburst, when little outburst material was present as a result of radiation pressure sweeping. By using Equations (5)–(6), we derived the absolute magnitudes of m_R(1, 1, 0) = 7.20 ± 0.10 (UT 2014 December 23), 8.82 ± 0.12 (UT 2015 January 11–13), and 5.61 ± 0.10 (UT 2015 January 16). With these magnitudes, the optical cross-sections of the dust cloud (C_c) were calculated by

$$\begin{eqnarray}&&{p}_{{\rm{R}}}\,{C}_{{\rm{c}}}=2.24\times {10}^{22}\,\pi \,{10}^{0.4({m}_{\odot }-{m}_{{\rm{R}}}(\mathrm{1,1,0}))},\end{eqnarray} \tag{ 7 }$$

where p_R is the geometric albedo (p_R = 0.04 was assumed), and m_⊙ = −27.11 is the R_C-band magnitude of the Sun at r_h = 1 au (Drilling & Landolt 2000).

By substituting ${m}_{{\rm{R}}}(1,1,0)$ in Equation (7), we obtained C_c = 3.32 × 10¹⁰ m² (UT 2014 December 23), 7.47 × 10⁹ m² (UT 2015 January 11–13), and 1.44 × 10¹¹ m² (UT 2015 January 16). We attributed the cross-section on UT 2015 January 11–13 to dust grains ejected via continual activity, which is irrelevant to these outbursts. We subtracted this value from the other two values and obtained the cross-section associated with the first outburst, C_c = 2.56 × 10¹⁰ m², and the second outburst C_c = 1.36 × 10¹¹ m². There is a possibility that we missed some fraction of the cross-section from the first outburst because our data might not cover its entire dust cloud on UT 2014 December 23. We thus considered the total cross-section of the first outburst ejecta as the lower limit (i.e., C_c ≥ 2.56 × 10¹⁰ m² for the first outburst ejecta).

The ejecta masses of the outbursts can be given by

$$\begin{eqnarray}&&{M}_{{\rm{d}}}=\displaystyle \frac{4}{3}{\rho }_{{\rm{d}}}{a}_{\mathrm{eff}}{C}_{{\rm{c}}},\end{eqnarray} \tag{ 8 }$$

where a_eff and ${\rho }_{{\rm{d}}}$ are the effective dust grain radius and mass density, respectively. The effective radius is given by

$$\begin{eqnarray}&&{a}_{\mathrm{eff}}=\displaystyle \frac{{\int }_{{a}_{\min }}^{{a}_{\max }}{a}^{3-q}{da}}{{\int }_{{a}_{\min }}^{{a}_{\max }}{a}^{2-q}{da}},\end{eqnarray} \tag{ 9 }$$

where q denotes the power index of the size distribution in the range of ${a}_{\min }\leqslant a\leqslant {a}_{\max }$.

With assumptions of ρ_d = 1000 kg m⁻³ and a_eff = 1 × 10⁻⁶ m (i.e., 1 μm), C_c gives M_d > 3.41 × 10⁷ kg for the first outburst and M_d = 1.81 × 10⁸ kg for the second outburst. These values are equivalent to masses of ${R}_{{\rm{c}}}\gt 20$ m and 35 m bodies. It should be noted, however, that this simple assumption likely underestimates the dust mass. Although the small particle (a_eff = 1 μm) assumption is occasionally quoted in previous research, abundant evidence exists for large particles from comets, which is subsequently discussed. Thus, it is better to consider that the masses should be the lowest limits. Ye et al. (2015) derived C_c = 7 × 10⁹ m² for the first outburst and C_c = 2 × 10¹⁰ m² for the second outburst, which are smaller than our estimates. In addition, they reported M_d = (2–3) × 10⁵ kg for the first outburst and M_d = (4–5) × 10⁵ kg for the second outburst, which are more than two orders of magnitude less than our estimates. Although we were unable to verify their results with the information available in their paper, we found that these orders-of-magnitude estimates reveal that the ejecta masses of 15P outbursts are equivalent to those of the 332P event (10⁸–10⁹ kg) (Ishiguro et al. 2014).

We further investigated the images soon after the second outburst. Figure 4 shows images taken after UT 2015 January 16.4 on the day in which brightening in Figure 2 was detected. We do not show the image taken on UT 2015 January 18 because of its bad quality (the total exposure time was only 3 minute due to unstable weather). To clarify the movement of dust grains via solar radiation pressure, we rotated these images to match the Sun–comet's direction to the horizontal direction. We produced color images, assigning g'-band images (the effective wavelength and the full width at half-maximum of λ_e = 483 nm and Δλ = 134 nm) to blue; R_C-band images (λ_e = 655 nm and Δλ = 121 nm) to green; and I_C-band images (λ_e = 799 nm and Δλ = 157 nm) to red colors. In these images, a whitish component expanded in space and simultaneously stretched toward the anti-solar direction. On the other hand, a bluish component expanded spherically with respect to the nuclear position. In general, prominent emissions associated with C₂ and CN appear in the g' band, weak emissions with NH₂ in the R_C-band, and negligibly faint signals with NH₂ and CN in the I_C-band (Brown et al. 1996; Meech & Svoren 2004, p. 317). In theory, dust particles are accelerated by solar radiation pressure, whereas neutral gas molecules are less sensitive to the radiation pressure and expand almost spherically. Thus, it is reasonable to consider that the whitish elongated structure originated from scattered sunlight from dust particles and that the bluish structure originated from CN and C₂ emissions.

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We attempted to extract the signals from the second outburst and to discriminate the gas flux from the scattered light by dust grains. The observed angles rotated very little over 2015 January 11–23 (that is, 04 for the phase angle, 24 for the position angles of the anti-solar vector, and 14 for the position angle of negative orbital velocity). Therefore, we subtracted the pre-outburst signals by using images observed on 2015 January 11 (IAO 1.05 m) and 2015 January 13 (OAO 0.5 m) from post-outburst images to adjust the apparent sizes of the comet based on the geocentric distances without rotating these images. Next, assuming that I_C-band images have little gas contamination, we subtracted them from g'-band images by adjusting the I_C-band intensity scales to minimize the extended dust structure (i.e., dust tail) in the residual images to produce gas intensity maps. Specifically, we detected a very weak spherical component in the I_C-band images that likely originated from weak CN emissions at 790–820 nm, which were noticeable until UT 2015 January 17 but were unclear after January 19. We subtracted the faint spherical component in the I_C-band images by using the above gas intensity maps, scaling the gas intensities to obscure the spherical components in the residuals to produce new dust intensity maps assuming that the intensity distributions of the gas components are the same in g' and I_C-bands.

The resultant images are shown in Figure 5. As expected, we extracted a spherical structure centered on the nucleus for the gas component associated with the second outburst (Figure 5 right). The surface brightness reduced rapidly in a few days not only because the gas diffused out in space, but also because the molecules had lifetimes equivalent to the observed period, i.e., two days for CN and five days for C₂ during the active solar phase (Huebner et al. 1992). We then examined the radius of the gas component from the images. Figure 6 shows the radial profile of the gaseous coma. Considering the sky background noise, we derived radii of 89 ± 5'' on UT 2015 January 16.4 and 182 ± 10'' on UT 2015 January 17.4. Assuming that the gas expanded at a constant speed since its ejection, we derived the onset time and expansion speed of the gas component as UT 2015 January 15.5 ± 0.2 and V_g = 1,110 ± 180 m s⁻¹ (Figure 7). We quoted the maximum ranges of these parameters as the errors in consideration of the margin of measurement because we had only two data points, which is insufficient for deriving the errors by using the least squares method. We found that the derived gas speed was slightly faster than the observed values of CN emission at other comets around 1 au, probably because we derived the speed at a very large distance from the nucleus (≈10⁸ km), where the gas flow velocity continued to increase (Krankowsky et al. 1986; Ip 1989).

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In the left-hand column of Figure 5, we show the dust component images. A short tail was visible on the first night (a), and an irregular cloud appeared on the second night (b) that extended toward the anti-solar direction via the radiation pressure. From the apparent cloud size, we derived the ejection onset time. We focused on the widths of the dust cloud perpendicular to the Sun–comet direction because dust particles were not accelerated by radiation pressure along that direction. Figure 7 shows the time evolution of the dust cloud width. The width increased linearly with the progression of time. By using a linear least squares method, we obtained the onset time on UT 2015 January 15.7 ± 0.1 for dust particles. Although this result agrees with the onset time of gas within the accuracy of our measurements, the trivial lag may suggest that the dust required more time to reach terminal velocity, which should be accelerated by the gas outflow. We conclude that the second outburst occurred on UT 2015 January 15.6–15.7 from gas and dust components. In addition, we derived the dust speed perpendicular to the Sun–comet direction as ${V}_{{\rm{d}}\perp }=280\pm 20$ m s⁻¹. Here, we should note that this speed might be misleading. Because we measured the speed perpendicular to the Sun–comet direction, we must underestimate the ejection speed for dust particles in ${V}_{{\rm{d}}\perp }$.

To derive the ejection speed, mass, and kinetic energy of the dust cloud in a comprehensive manner, we conducted a model simulation to reproduce the observed morphologies of dust ejecta from the second outburst by following the scheme in Ishiguro et al. (2013). The morphologies of dust clouds are generally determined by the ejection speeds and the effects of solar radiation pressure, which can be approximately given by a function of grain size. Therefore, the sizes and ejection speeds of the dust particles can be determined essentially through an investigation of dust cloud morphologies. For a spherical particle, the ratio of the radiation pressure with respect to the solar gravity (β) is given as β = $5.7\times {10}^{-4}\,{Q}_{{\rm{pr}}}/{\rho }_{{\rm{d}}}a$, where a and ${\rho }_{{\rm{d}}}$ are dust radius (m) and mass density (kg m⁻³), and Q_pr is a radiation pressure coefficient (Finson & Probstein 1968; Burns et al. 1979). Here, we assumed ρ = 1 × 10³ kg m⁻³ and Q_pr = 1. For convenience of fitting, we assumed that the dust ejecta consisted of two components, a high-speed envelope and a low-speed tail. We set the ejection epoch of UT 2015 January 15.5, which has negligible influence on the result within the error range (i.e., 0.1 day). For simplicity, we assumed that dust particles were ejected symmetrically with respect to the solar direction within a cone of a half opening angle of w. We employed a size-dependent ejection speed of V_ej = ${V}_{0}{a}^{k}$ and power-law size frequency distribution given by N(a) = ${N}_{0}{a}^{-q}$ in the range between a_min and a_max. Because the ranges of these parameters and fitting scheme are the same as those given in Ishiguro et al. (2013), we do not describe the details here.

Through the fitting, we derived the best-fit parameters as shown in Table 2. Because large particles are not sensitive to solar radiation pressure and still reside near the nucleus, we found it impossible to derive the maximum size of particles. We fixed the minimum value for β to 6 × 10⁻⁴ (i.e., a = 1 mm) on the grounds that large cometary boulders may not be ejected efficiently through outbursts (Bertini et al. 2015). Figure 8 shows a comparison of the observed dust cloud morphology and our best-fit model, which broadly match. However, the entire morphology was not perfectly reconstructed by our model, probably because the dust ejection was not symmetric with respect to the solar direction. Although we do not intend to upgrade the model fitting by fine tuning the central axis of the dust emission here, we would insist that the outburst location might deviate slightly from the sub-solar point because of this asymmetry. The maximum β was determined well to be β_max = 1.6 ± 0.2 because our observation data covered the faint end of the dust cloud, where particles with ${\beta }_{\max }$ were dominant. To fit the width and sunward extension of the cloud, we obtained the best fit parameters V₀ = 450 ± 30 m s⁻¹ for the envelope particles and V₀ = ${330}_{-30}^{+60}$ m s⁻¹ for the tail. These parameters resulted in a maximum speed for the smallest particles of V_max = 570 ± 40 m s⁻¹. For confirmation, we calculated the speed perpendicular to the Sun–comet direction with these simulation results, ${V}_{{\rm{d}}\perp }$ = ${V}_{\max }\,\sin (w)$ = 285 ± 40 m s⁻¹, which is consistent with the value given in Section 3.5 (i.e., ${V}_{{\rm{d}}\perp }$ = 280 ± 20 m s⁻¹). From the model fitting, we determined that ∼20% of the cross-section originated from the envelope, whereas ∼80% was from the tail and coma. We calculated the total mass and kinetic energy for the second outburst by using the parameters derived from the simulation (see Table 2) and obtained M_d = 7.0 × 10⁸ kg and E_d = 6.5 × 10¹² J for the second outburst.

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Table 2.  Dust Model Parameters

Parameter	Input Values	Best-fit (Envelope)	Best-fit (Tail + Coma)	Unit
u1	0.1–0.9 with 0.1 interval	0.4 ± 0.3	0.55 ± 0.1	⋯
q	3.0–4.5 with 0.1 interval	3.8 ± 0.1	3.8 ± 0.1	⋯
${\beta }_{\max }$	1.0–2.5 with 0.1 interval	1.6	1.6	⋯
${\beta }_{\min }$	0.5, 0.3, 0.1, 0.01, 0.001	0.3	6 × 10−4 (fixed)	⋯
V0	150–600 with 30 interval	450 ± 30	${330}_{-30}^{+60}$	m s−1
${\sigma }_{v}$	0–0.5 with 0.1 interval	0.2 ± 0.1	0.4 ± 0.1	⋯
ω	5–60 with 5 interval	35 ± 10	30 ± 10	degree

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Table 3.  Comparison of 15P, 322P, and 17P Outbursts

Quantity	15P	332P	17P	References
aa	3.488	3.090	3.621	(1)
eb	0.720	0.489	0.432	(1)
ic	6.799	9.378	19.090	(1)
qpd	0.976	1.579	2.057	(1)
TJ	2.620	3.010	2.859	(1)
rhe	0.99, 1.02	1.59	2.44	(2), (3)
RNf	0.9	<1.9	2.1	(4), (2), (5)
${\rm{\Delta }}{t}_{p}$ g	−11, +19	+20	+172	(3)
${m}_{R}(1,1,0)$ h	7.2, 5.6	6.0	−1.1	(2)
Cci	>2.6 × 1010, 1.4 × 1011	1.0 × 1011	7.1 × 1013	(2), (6)
${t}_{\mathrm{rise}}$ j	<3	$\approx 1$	1.2 ± 0.3	(7)
${t}_{\mathrm{fade}}$ k	⋯	70	50	(8)
Mdl	>2 × 108, 7 × 108	5 × 108	(4–8) × 1011
Vmaxm	570 ± 40	500 ± 40	554 ± 5	(2),(10)
Ekn	6.5 × 1012 for 2nd	5.0 × 1012	(1–6) × 1015	(2), (10)
Ek/Mdo	1 × 104	1 × 104	104–105	(2), (7), (10)

Notes.

^aSemimajor axis in au. ^bEccentricity. ^cInclination in degree. ^dPerihelion distance in au. ^eHeliocentric distance at the time of outburst in au. ^fRadius of nucleus in km. ^gOnset time after perihelion passage in days. ^hAbsolute R_C-band magnitude. ⁱTotal cross-section of dust cloud in m². ^jRise time in days. ^kFade time when the magnitude decreased by 4 mag in days. ^lEjecta mass in kg. ^mMaximum speed of ejecta in m s⁻¹. ⁿKinetic energy in J. ^oKinetic energy per unit mass in J kg⁻¹.

References. (1) ssd.jpl.nasa.gov, (2) Ishiguro et al. (2014), (3) Hsieh et al. (2010), (4) Fernández et al. (2013), (5) Stevenson et al. (2014), (6) Ishiguro et al. (2013), (7) Li et al. (2011), (8) Stevenson & Jewitt (2012), (9) Lin et al. (2009), (10) Ishiguro et al. (2015).

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Cometary outbursts provide opportunities to investigate fresh cometary materials that are embedded in the surface processed layers but appear when a large amount of material is ejected via explosions. In particular, the 15P event in 2015 January provided a unique opportunity to determine the maximum β value with good precision for several reasons. Because we determined the precise onset time of the outburst, it was easy to follow the motion with respect to the reference time. In addition, the outburst ejecta were observed at a large phase angle of α = 45° (c.f., α = 19° for 332P and 17° for 17P), which enabled observation of the dust motion toward the anti-solar direction more clearly. Moreover, the outburst occurred at a smaller heliocentric distance (1.0 au), where the acceleration by radiation pressure was detected more effectively than those at distant locations such as 1.6 au for 332P and 2.4 au for 17P.

It is well known that radiation pressure force depends on not only size but also composition. The β values have been investigated theoretically considering fluffy dust particles (Mukai et al. 1992) and a variety of compositions (Burns et al. 1979). In Section 3.6, we adopted a simple model in which β is inversely proportional to size under an assumption of a radiation pressure coefficient of Q_pr = 1. However, this assumption holds only when the particle size is larger than the optical wavelength (λ = 0.5 μm). Q_pr has almost constant value for a > 0.1–0.3 μm and significantly drops as the size of dust grains decreases (Ishiguro et al. 2007). For fluffy dust particles, β is less dependent on the aggregate size but similar to that of each constituent (Mukai et al. 1992). Wilck & Mann (1996) calculated the β values of dust particles by using a core–mantle spherical particle model composed of silicate and amorphous carbon with different porosities, and suggested that cometary dust particles have β_max = 1–1.8. Transparent materials such as silicates and water ice tend to have small β maximum values (i.e., β_max < 1), whereas that of absorbing particles such as carbon is β_max > 1 (see e.g., Kimura et al. 2016). The β_max value determined in our measurement, 1.6 ± 0.2, is consistent with that in the porous absorbing particles model (Wilck & Mann 1996) and silicate-core, organic-coated grains model for dust aggregates (Kimura et al. 2003). However, our value is inconsistent with those of silicate spheres (β_max < 1) and organic spheres (β_max > 3).

Cometary outbursts have been observed in a wide variety of comets. There are some references that analyzed historical observations of cometary outbursts (e.g., Hughes 1990). However, the occurrence frequency and mass production rate have not been studied well because there is no coordinated observation system for monitoring comet magnitude. Some outbursts could have been missed because the magnitude contrast was too weak before and after to be noticed as an outburst or because the observation condition was not suitable for detection (i.e., too close to the Sun or Moon). Nevertheless, some outburst events have been reported in a voluntary manner by amateur observers. For example, the 17P event was first reported by A. Henriquez Santana (Spain), who noted sudden brightening to 8.4 mag by using a 0.2 m reflector (Buzzi et al. 2007). 332P was discovered in Japan during its outburst by S. Murakami and K. Ikeya. They observed this event by looking into the eyepieces of 0.46 and 0.25 m telescopes when it reached ∼9 mag (Murakami 2010). Moreover, 15P double outbursts with brightening to 8–9.4 mag were also reported by amateurs through a mailing list of comet observers (Ye et al. 2015).

It has been suggested that 17P and 332P events were caused by a phase change of amorphous water ice (Li et al. 2011; Ishiguro et al. 2013), although other mechanisms such as rotational breakup of brittle cometary nuclei cannot be ruled out (Li & Jewitt 2015). Excavations via impacts might create environments favorable to outbursts, as suggested in Beech & Gauer (2002). Because the energy per unit mass is similar to both events, we conjecture that 15P events were also caused by a phase change of amorphous water ice. Here, we consider the frequency of such outbursts in the inner (${r}_{\odot }\lesssim 5$ au) solar system, where the physical mechanism of general cometary activity is confined to sublimation of water ice rather than that of super volatiles such as CO (Jewitt 2009). For this reason, we excluded frequent outbursts at 29P/Schwassmann-Wachmann 1 beyond the Jupiter orbit. Moreover, we restricted our discussion to JFCs and Encke-type comets (ETCs) because significant observation samples are not available for Halley-type comets (HTCs) and long-period comets. We applied a sample cut of the perihelion distance $q\lesssim 5\,\mathrm{au}$ to the JPL Small-Body Database Search Engine¹⁴ and retrieved 505 JFCs and ETCs in the list of known comets as of the end of 2015. We regarded disintegrated comets as a single object (e.g., 73P/Schwassmann-Wachmann 3 B, C, G... constituted a united body). Moreover, we excluded disappeared or dead comets (3D/Biela and 5D/Brorsen), designated as "D/," and main-belt comets, which are unlikely to contain amorphous ice (Prialnik & Rosenberg 2009). Among JFCs and ETCs, 357 objects passed their perihelion at least once between 2007 October and 2015 December, which make up ∼70% of the entire population. This means that ∼70% of JFCs and ETCs might have a chance of displaying outburst activities owing to the additional heat from the Sun near their perihelia.

We attempted to find outburst events through the Smithsonian Astrophysical Observatory/National Aeronautics and Space Administration (SAO/NASA) Astrophysics Data System (ADS) Astronomy Query Form by inputting "comet" and "outburst" as abstract terms and manually plumbing the results by reading the abstracts. In addition, we added one event at 205P which had an outburst showing similar appearance to 15P (S. Yoshida 2016, private communication). We found that 15 outburst events occurred at 11 comets since 2007 (Table 4). We chose the arbitrary period beginning in 2007 because researchers have increased their consciousness toward outbursts since then, as motivated by the 17P event.

Table 4.  List of Outbursts at JFCs and ETCs since 2007 October

Name	Date	rh	Δ	α	${m}_{-}$ a	${m}_{+}$ b	${m}_{-}(1,1,0)$ c	${m}_{+}(1,1,0)$ d	Cce	References
205P/Giacobini	2015 Sep 26	1.92	1.57	32	∼20	∼14	16.5	10.5	$1.6\times {10}^{9}$	(1)
17P/Holmes	2015 Jan 26	2.99	2.38	26	19.2	16.8	14.0	11.6	$5.1\times {10}^{8}$	(2)
15P/Finlay	2015 Jan 15	1.02	1.39	45	11.1	8.0	11.1	5.6	1.4 $\times {10}^{11}$	(3)
	2015 Jan 7	0.99	1.40	45	13.2	12.7	10.9	10.4	$6.8\times {10}^{8}$	(3)
	2014 Dec	0.99	1.48	41	⋯	9.4	⋯	7.2	$2.6\times {10}^{10}$	(3)
63P/Wild 1	2013 May 16	1.98	1.58	30	15.6	13.5	12.0	10.0	$2.2\times {10}^{9}$	(4)
168P/Hergenrother	2012 Aug–Dec	1.42	0.43	13	15.2	8.0	15.8	8.6	$9.1\times {10}^{9}$	(5)
P/2010 V1	2010 Nov 2	1.59	2.34	19	⋯	9.5	⋯	6.0	$1.0\times {10}^{11}$	(6)
103P/Hartley 2	2010 Aug–Nov	1.06–1.56	0.12–0.70	29–59	⋯	⋯	⋯	⋯	⋯	(7)
81P/Wild 2	2010 Apr–Aug	1.70–2.21	0.70–1.85	4.5–27.1	⋯	⋯	⋯	⋯	$2.2\times {10}^{9}$	(8)
P/2010 H2	2010 Apr	3.11	2.13	5	>20	12.6	15.7	8.3	$1.2\times {10}^{10}$	(9)
217P/LINEAR	2009 Oct	1.31	0.61	47	11.3	9.3	10.1	8.1	$1.2\times {10}^{10}$	(10)
199P/Shoemaker 4	2008 Aug 3	3.41	3.35	17	17.8	14.5	14.1	8.6	$8.9\times {10}^{9}$	(11)
6P/d'Arrest	2008 Aug	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	(11)
17P/Holmes	2008 Jan 5	4.18	3.28	6	19.8	19.3	13.9	13.4	$4.5\times {10}^{7}$	(12)
	2007 Nov 12	2.51	1.62	13	12.8	12.6	9.2	9.0	$1.1\times {10}^{9}$	(13)
	2007 Oct 23	2.44	1.64	17	17	2.5	13.4	−1.1	$7.1\times {10}^{13}$	(14)

Notes.

^aPre-outburst apparent magnitudes. ^bPost-outburst apparent magnitudes. ^cPre-outburst absolute magnitudes calculated with Equation (5). ^dPost-outburst absolute magnitudes calculated with Equation (5). ^eCross-section calculated with Equation (7) assuming the geometric albedo p_R = 0.04.

References: (1) S. Yoshida (2016, private communication), (2) Kwon et al. (2016), (3) This work, (4) Opitom et al. (2013), (5) Sekanina (2014), (6) Ishiguro et al. (2013), (7) Milani et al. (2013), (8) Bertini et al. (2012), (9) Vales et al. (2010), (10) Sarugaku et al. (2010), (11) Miles 2009, (12) Miles (2010), (13) Stevenson & Jewitt (2012), (14) Li et al. (2011).

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Table 4 shows some minor outbursts detected by observers with larger telescopes because these comets drew their attention due to space mission targets (81P/Wild 2 and 103P/Hartley 2) and repeated outbursts at 17P and 15P. Figure 9 shows the cumulative mass distribution of the outburst ejecta. We note that six outburst events, including 17P, 168P/Hergenrother, 332P, 217P/LINEAR, and two events at 15P, were brightened down to 10 mag. Such events would be detectable with observation through inexpensive equipment such as ∼10 cm class telescopes. Thus, we conjecture that outbursts $\lesssim 10$ mag would be detected almost completely if observed conditions allowed ground-based observers to make observations.

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We fitted the ejecta mass distribution with a power-law function for five objects (i.e., $\lesssim 10$ mag events except 17P) and obtained a power index of γ = 0.45 ± 0.09. Although there might be no physical basis to support the power-law function in Figure 9, it is likely that the 17P outburst in 2007 was a very rare phenomenon to be detected within ∼eight years because the occurrence is one order of magnitude higher than that expected by the power function. In fact, the large discrepancy for 17P must be a bias of our analysis in which we counted the number of outbursts since the epoch-making event. In addition, the power-law function fits well in the mass range of 6 × 10⁷–1 × 10⁹ kg but deviates from the observed values in the mass range of ≲6 × 10⁷ kg. Some outbursts might not be noticed because of the faintness or weak contrast before and after outbursts. For these reasons, we consider that the unbiased outbursts rate in which the ejecta mass corresponds to 6 × 10⁷–1 × 10⁹, (hereafter referred to as "15P-class" outbursts).

We applied the Poisson distribution following Sonnett et al. (2011):

$$\begin{eqnarray}&&P(n)=\displaystyle \frac{{({fCN})}^{n}\exp (-{fCN})}{n!},\end{eqnarray} \tag{ 10 }$$

where n and f denote the number of outburst detections and incidence of outbursts, N is the number of observed samples, and C (∈ [0, 1]) is the completeness of the survey. The 90% upper confidence limit on the incidence, ${f}_{90 \% }$, is given by

$$\begin{eqnarray}&&0.9=\displaystyle \frac{{\int }_{0}^{{f}_{90 \% }}P(n){df}}{{\int }_{0}^{1}P(n){df}}.\end{eqnarray} \tag{ 11 }$$

Assuming that observers could detect 15P-class outbursts completely at the solar elongation of ${\epsilon }_{\odot }\,\gt$ 60°, we have C = 0.67. Over eight years since October 2007, there were n = 7 15P-class outbursts out of N = 357 comets that passed their perihelion. By solving the implicit Equation (11), we obtained f_90% = 0.05, which suggests an expected number $\langle n\rangle$ = ${f}_{90 \% }{CN}$ ∼ 12 in eight years, or 1.5 times per year. Therefore, we can express the unbiased cumulative and differential frequency of 15P-class outbursts f_ub and ${f}_{{\rm{ub}}}^{\prime }$ as

$$\begin{eqnarray}&&{f}_{{\rm{ub}}}(\gt M)={\int }_{M}^{\infty }f{^{\prime} }_{{\rm{ub}}}(m)\,{dm}=A{\left(\displaystyle \frac{M}{{M}_{0}}\right)}^{-\gamma },\end{eqnarray} \tag{ 12 }$$

where A = 1.5 yr⁻¹ and M₀ = 6 × 10⁷ kg are constants. We obtained an effective mass production rate of 15P-class outbursts of ∼3 × 10⁸ kg year⁻¹ or ∼10 kg s⁻¹, from (see Appendix)

$$\begin{eqnarray}&&\langle m\rangle ={\int }_{{M}_{1}}^{{M}_{2}}{{mf}}_{{\rm{ub}}}^{\prime }(m)\,{dm},\end{eqnarray} \tag{ 13 }$$

where M₁ = 6 × 10⁷ kg and M₂ = 1 × 10⁹ kg are lower and upper bounds on the power-law behavior of 15P-class outbursts, respectively. We consider that the incidence would be underestimated because we optimistically assumed C = 0.67. Some of the outbursts were missed owing to the crowded region of stars or the full lunar phase. Even considering these factors, which may decrease C by several factors, our result would obtain one order-of-magnitude accuracy for the unbiased frequency. Therefore, it is reasonable to think that 15P-class outbursts inject dust particles into the interplanetary space at a rate of $\gtrsim 10\,\mathrm{kg}$ s⁻¹.

The mass is ∼2 orders of magnitude less than the mass required to sustain the interplanetary dust cloud. This implies that 15P-class outburst events may not compensate the mass eroded by Poynting–Robertson drag and other dynamical mechanisms onto the interplanetary dust. However, if we integrate ${{mf}}_{{\rm{ub}}}^{\prime }(m){dm}$ up to the 17P-class ejecta mass (i.e., M₂ = 1 × 10¹² kg), the mass production rate from 15P–17P class events would be ∼500 kg s⁻¹, although we are not sure whether the frequency follows a simple power-law function up to the 17P class. These results suggest that large-scale cometary outbursts might contribute a significant fraction of the interplanetary dust source.