HABITABILITY OF EXOMOONS AT THE HILL OR TIDAL LOCKING RADIUS

In order to take into account multiple sources of radiation, Heller & Barnes (2013) developed a model that determines the effect of stellar illumination, light from the star reflected off the planet, the planet's thermal radiation, and tidal heating on the surface of an exomoon. Their code allowed for the specification of a variety of system parameters, such as stellar host-mass (M_s), planet mass (M_p), moon mass (M_m), star radius (R_s), planet radius (R_p), stellar effective temperature (T_eff), Bond albedo of the planet (α_p), rock-to-mass fraction of the moon (rmf), semi-major axis of the planet to the star (a_sp), eccentricity of the planet (e_sp), semi-major axis of the moon about the planet (a_pm), eccentricity of the moon's orbit (e_pm), inclination of the moon with respect to the planet (i), and orientation of the inclined orbit with respect to the periastron (ω). With these prescriptions, they were able to calculate the total flux received on the moon during one moon orbital phase (or "phase curves"), the average flux with respect to the moon's latitude and longitude after one complete planet revolution (or "flux map"), as well as the physical orbit of the planet and moon as they orbit the host-star (or "orbital path").

While the Heller & Barnes (2013) code was very extensive and thorough, there are some limitations to the possible applications. For example, the authors assumed that the heat from gravitationally induced shrinking of the gaseous planet was negligible compared to the stellar luminosity. In a similar vein, the ratio of the period of the moon about the planet (P_pm) to the period of the planet about the star (P_sp) follows P_pm/P_sp ≲ 1/9, as per Kipping (2009)—see Section 2.3 for more discussion. Also, the moon is assumed to be in a relatively circular orbit due to the effects of tidal heating dampening the orbital eccentricities, unless the moon is perturbed by interactions with other bodies. In their paper, Heller & Barnes (2013) explored e_pm ⩽ 0.05. Finally, all moons are assumed to be tidally locked, the moon never experiences a penumbra from the planet, M_p ≫ M_m, and both atmospheric and geologic contributions are ignored. We suggest the reader consults the original paper for a more thorough description of the techniques and simplifications employed by Heller & Barnes (2013).

Using their publicly available code (see their Appendix C), we were able to adjust their methodology to calculate the average flux experienced on the moon's surface (across all latitudes and longitudes) over the course of one planet revolution. The flux experienced on the moon was calculated assuming blackbody radiation emitted by the star. In Figure 1 we show a year-averaged flux profile for an example system featuring an Earth-sized moon orbiting a Jupiter-sized planet, at 1 AU from the Sun. The different color lines show heat contributions from the star (orange), planetary reflection (blue), and planetary thermal emission (red) which are summed to the total (purple). We have split the figure into two flux-range regimes in order to better illustrate the contribution from all three sources, while noting that all three components (stellar, reflective, and thermal) lead to the total. The average total flux oscillation experienced on the moon as a result of its orbit around the planet is given in the top left corner of the plot. Details of this example system are found in Table 1, such that the eccentricity of the planet e_sp = 0.0. The inclination of the moon is i = 0° and the planet–moon semi-major axis is 0.01 AU. The moon's orbit has an eccentricity of e_pm = 0.0, meaning that tidal heating has no effect on the flux of the moon. The total flux of the moon in Figure 1 is dominated by the stellar luminosity, with smaller oscillations due to the revolution of the moon about the planet, as can be seen in the light reflected off the planet.

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As the number of confirmed exoplanets increases, so does the ability to characterize the systems and determine widespread patterns. The eccentricity of the host-planet has a direct consequence on the flux on the exomoon, since it affects the distance between the host-star and the planet–moon system, as studied by Heller (2012). Figure 2 (left) shows all of the confirmed radial velocity (RV) exoplanets with measurements for both planet mass and eccentricity, totaling 441 planets. Overlaid on the plot are the median (red) and mean (green) masses as determined for 0.1 eccentricity bins. The median within a bin rules out outlier masses and the mean determines the average per bin. Both trends indicate that as eccentricity increases until e_sp ∼ 0.7, planet mass increases. For e_sp > 0.7, planet mass tends to decrease. The RV technique is biased toward the lower-eccentricity planets, such that the number of planets decreases with higher eccentricities. In addition, there is also a bias in observing higher mass planets. Despite these idiosyncrasies in the data, we wish to study what is currently considered a "typical" system in the high-eccentricity regime, bearing in mind that this definition may change in the future.

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We continue with the example system in Figure 2 (right), where we have changed the eccentricity, e_sp = 0.4, which is on par for a more massive planet. The high eccentricity dominates the moon's flux phase profile such that it is clear when the planet–moon system is at periastron (planetary phase = 0.0) and when it is at apastron (planetary phase = 0.5). Due to the close proximity of the planet–moon to the host-star, not only does the moon's flux start +500 W m⁻² higher than at low eccentricity (see Figure 1), but the average total small-scale flux variation is higher by 28.08 W m⁻². On the other hand, the average flux on the moon at low eccentricity is 262 W m⁻² while at high eccentricity it is 283 W m⁻², so not significantly different.

The final system variable that has a large impact on the flux on the moon's surface is the distance between the planet and the moon, where the semi-major axis is a_pm. As we have seen in the previous two incarnations of our example system, a moon that is relatively close to the planet experiences somewhat small flux fluctuations as a result of its close orbit. Since there have not been any observations of exomoons to-date, we wish to explore the limits of gravitational influence on the moon.

We first consider the radius at which a moon is gravitationally bound to a planet, due to the influence of the nearby host-star, or the Hill radius:

$$\begin{equation} r_H = a_{sp}\, \chi \,(1-e_{sp})\, \sqrt [3] {\frac{M_p}{3\,M_s}}. \end{equation} \tag{ 1 }$$

Here χ is an observational factor implemented to take into account the fact that the Hill radius is just an estimate and that other effects may impact the gravitational stability of the system. Following both Barnes & O'Brien (2002) and Kipping (2009), a conservative estimate is that $\chi \mathrel \lesssim 1/3$. We choose to use χ ∼ 1/3 such that $P_{pm} / P_{sp} \mathrel \lesssim$ 1/9 (Domingos et al. 2006; Kipping 2009). This means that a moon beyond this cautious radius would be perturbed by tidal interaction with the star and would not be able to maintain a stable orbit around the planet.

Second, we look at the distance at which a planet's tidal influence induces the moon to become locked in a synchronous rotation. The tidal locking radius is defined as:

$$\begin{equation} r_{TL} \approx \left (\frac{3\, {G M}^2_p\, k_2\, R^5_m \,t_L}{\omega _{0}\, I \, Q}\right)^{1/6}, \end{equation} \tag{ 2 }$$

where G is the gravitational constant, k₂ = 0.3 is the second-order Love number, t_L is the timescale for the moon to become locked, ω₀ is the initial spin rate of the moon in radians per second, $I \approx 0.4\,M_m\,R^2_m$ is the moon's moment of inertia, and Q is the dissipation function of the moon (Peale 1977; Gladman et al. 1996). From Kasting et al. (1993), we take ω₀ = one rotation per 13.5 hr and Q = 100 for a conservative estimate. Based on the present age of the Earth, t_L = 4.5 Gyr.

In Figure 3 (left) we analyze the Hill and tidal locking radii with respect to planet mass in a log–log plot. The solid lines show the Hill radii for a variety of stellar masses, while the dashed lines give the tidal locking radius when t_L = 4.5 Gyr (black) and 4.5 Myr (red). As planetary mass increases, so does the distance at which gravity keeps the moon both bound and tidally locked. We will be using the typically accepted value t_L = 4.5 Gyr, but find it interesting to consider the range of values produced by altering this variable, especially with respect to the Hill radius, for different stellar masses. For all solar and planet masses at a distance of 1 AU, particularly the 1.0 M_☉ star and 1.0 M_Jup planet in our example system, the Hill radius is smaller than the tidal locking radius when t_L = 4.5 Gyr.

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Revisiting our example system, we have plotted the flux profile with respect to planetary phase in Figure 3 (right). However, in this instance we have placed the planet near the Hill radius, r_H = 0.022 AU, where a_pm = 0.02 AU. For this system, the orbit of the moon dominates the flux profile such that P_pm/P_sp = 1/10, which is within the bound determined by Kipping (2009) and Heller & Barnes (2013). The average total flux fluctuation, ∼43.07 W m⁻², is smaller in this example system compared to the same system at various planetary eccentricities, most likely as a result of the larger a_pm. The average total flux is also similar to the previous incarnations of the example system at 262 W m⁻².

We have analyzed the hypothetical exomoon in three scenarios that only vary slightly from one another. When the eccentricity of the planet–moon system is large, the surface of the exomoon experiences much more extreme flux variations compared to a circular orbit. The average flux of the exomoon at a large eccentricity is also slightly larger at high eccentricity compared to e = 0. Comparatively, when the exomoon is placed at a distance further from the planet in accordance with the Hill (in this case) or tidal radius, the average flux on the surface is the same as the original scenario. In other words, the thermal and reflected radiation from the planet do not make a significant contribution to the flux range experienced on the moon, while the eccentricity of the planet–moon system results in an extreme range of fluxes and slightly higher average flux.

The μ Ara system contains four confirmed exoplanets around a 1.15 M_☉ G3IV-V type star (Pepe et al. 2007), see Figure 4 (top) where the d-planet is too close to the host-star to be distinguished. While the c-planet is the most massive at 1.89 M_Jup, it is also the furthest away such that a_sp = 5.34 AU. The 1.746 M_Jup b-planet is the only planet to be within the habitable zone (HZ), or between [1.22, 2.14] AU, for the entirety of its orbit, where a_sp = 1.527 AU and e_sp = 0.128. We chose to study the μ Ara system because of the presence of numerous, relatively massive, planets. The large mass of the b-planet results in a large Hill sphere which presents many possibilities for a moon system, both in terms of masses and moon–planet separations. In addition, the eccentricity of the b-planet is enough such that it may vary the equilibrium temperature of the planet–moon system by a substantial amount, while remaining fully within the habitable zone.

In Figure 5 (left) we explore the maximum distance that a hypothetical 0.25 M_⊕ moon could orbit stably and remain tidally locked. From the plot, the tidal locking radius is more dominant in this system, making the Hill radius for a near solar-mass star the limiting distance for the placement of the hypothetical moon. The Hill radius for μ Ara b is r_H = 0.035 AU; therefore, to ensure the planet is fully within the Hill sphere, a_pm = 0.03 AU.

The flux profile for the hypothetical moon near the Hill radius is shown in Figure 5 (right). Similar to Figure 3, the large orbital period of the moon has a strong influence on the flux phase fluctuations. However, the non-zero eccentricity of the planet–moon system is also apparent in the flux profile, much like Figure 2 but not as exaggerated. The moon's surface experiences a minimum flux of 130 W m⁻² and a maximum of 256 W m⁻², with the average total flux oscillating ∼37 W m⁻² as the moon's phase changes.

While the HD 28185 system only contains one confirmed exoplanet (Minniti et al. 2009), that planet is 5.8 M_Jup  and maintains an orbit that is fully within the habitable zone at a_sp = 1.023 AU and e_sp = 0.05 (Table 1). Given the mass extent of the confirmed exoplanet, we wanted to explore the theoretical moon–planet interaction within the enormous Hill sphere. However, per one of the four simplifications noted in Heller & Barnes (2013), we had to ensure that the distance between the moon and the planet was not so large that P_pm ≈ P_sp. Therefore, a system with a large planet had to be found that also fell nearer the inner habitable zone boundary. In contrast to μ Ara, HD 28185 b has a very low eccentricity, such that the flux received on the moon will be affected the most by the distance of the planet from the moon. The habitable zone around the host-star, a G5V type star with a mass of 0.990 M_☉, extends from 0.95 to 1.66 AU (Figure 4, second from the top).

Figure 6 (left) is similar to Figure 5 (left) in that it examines the distance limit for a hypothetical moon as defined by either the Hill radius or the tidal locking radius for the system. Like μ Ara, the gravitational boundary for HD 28185 b is more restricted by the Hill radius for a 1.0 M_☉ star as compared to the tidal locking radius at t_L = 4.5 Gyr. The Hill radius is r_H = 0.039 AU, where we define a_pm = 0.035 AU in order to guarantee the moon is fully within the Hill radius.

In Figure 6 (right) we show the flux profile for the hypothetical moon around HD 28185 b. Because of the relatively low planetary eccentricity, the moon's orbit dominates the flux fluctuations seen on the surface. In this scenario, P_pm/P_sp ∼ 1/11. The exoplanet orbiting HD 28185 is one of the largest planets that spends the entirety of its orbit within the habitable zone. Yet, despite its large size, the thermal and reflected flux contributions, ∼0.01 W m⁻² and ∼0.05 W m⁻², respectively, from this planet are nearly identical to that from μ Ara b. Therefore, the larger average flux variations seen on the moon's surface, ∼56 W m⁻², are most likely due to the larger semi-major axis between the moon and the planet.

Unlike the previous applications, we wanted to analyze a system that did not spend the entirety of its period within the habitable zone. We chose to analyze the BD +14 4559 system because the planet has a highly eccentric orbit, e_sp = 0.29, and only spends 68.5% of its orbital phase within the habitable zone (Figure 4, second from the bottom). The single confirmed exoplanet orbiting BD +14 4559 has a mass that is 1.52 M_Jup (Niedzielski et al. 2009), which places it between both the median and mean lines in Figure 2 (left). Taking into account the observational bias in the RV technique, which preferentially detects planets with small eccentricity, we expect this to be a standard eccentricity for an RV-observed planet of this mass. In this way, we could explore the flux received on the exomoon in a "standard" high-eccentricity system, despite the host-planet being far from the host-star for a large fraction of its orbit. The observed b-planet has an orbit at 0.776 AU from the 0.86 M_☉ K2V type star (Table 1), where the boundaries of the habitable zone fall between 0.52 and 0.94 AU.

Due to the higher eccentricity, the distance to the Hill radius is smaller than those seen in the other applications (Figure 7, left). In contrast, the tidal locking radius has not shifted significantly, making the two radii even more discrepant. Similar to both μ Ara b and HD 28185 b, the hypothetical moon orbiting the b-planet is most limited by the Hill radius, r_H = 0.015 AU. For our purposes, we place the hypothetical 0.25 M_⊕ moon at a distance of a_pm = 0.01 AU.

The closer proximity of the hypothetical moon to the planet means that P_pm/P_sp ∼ 1/27, as shown in Figure 7 (right). The shorter moon orbital period is most likely the cause of the smaller average total flux fluctuation, ∼23 W m⁻². As a result, the relatively large planetary eccentricity dominates the flux profile, which swings from 241 W m⁻² to 63 W m⁻².

Out of the +440 confirmed RV planets with planet mass and eccentricity measurements, only 35 of them had a Hill radius at a further distance than the tidal locking radius, using our moon with 10 × the mass of Ganymede = 0.25 M_⊕ and t_L = 4.5 Gyr. Comparatively, when the same test was run for a moon with M_m = 10 M_⊕, there were 44 planets with a smaller tidal locking radius. Out of the 35 super-Ganymede-moon systems, only 3 of them spent any part of their orbit within the habitable zone: HD 4732 c (70%), HD 73534 b (62.7%), and HD 106270 b (20.7%). The habitable zone boundaries for HD 4732 c were [4.18, 7.49] AU, which we believed placed the exoplanet too far from the host-star such that the heat from gravitationally induced shrinking of the gaseous planet was no longer negligible. Similarly, HD 106270 b spent such a small percentage of its orbit in the habitable zone that we felt the planet and moon would be too cold. Therefore, we have investigated HD 73534 b (Valenti et al. 2009), which is a 1.104 M_Jup planet orbiting a 1.23 M_☉ G5IV type star at a_sp = 3.068 AU and e_sp = 0.074 (Figure 4, bottom).

Figure 8 (left) confirms that the tidal locking radius with t_L = 4.5 Gyr is smaller than the Hill radius for a solar-type star, 0.055 AU and 0.062 AU, respectively. Therefore, we have placed the hypothetical moon at a distance of a_pm = 0.05 AU from the host-planet, to ensure that the moon is well within the tidal locking limits.

The flux profile for HD 73534 b is shown in Figure 8 (right), where P_pm/P_sp = 1/13, due to the large planet–moon semi-major axis. However, while the a_pm for this system is the largest that we have investigated, it is tempered by the fact that a_sp = 3.068 AU, which is also the largest planetary semi-major axis within our applications. Therefore, we do not see the large average total flux fluctuation on the surface of the moon, ∼15 W m⁻², as seen in Figure 6 (right). The maximum flux experienced by the moon's surface in this system is 108 W m⁻², while the minimum is 69 W m⁻².

It could be said with some confidence that the incident flux gradient from the equator to the poles of the moon results in significantly lower equilibrium temperature at the pole. This idea holds regardless of the invoked climate model, especially given that we defined the incident angle between the moon and the planet as i = 0° for all system applications. However, the complicated, multi-parameter climate model's dependencies make it necessary to use the incident flux as a first-order approximation for the thermal equilibrium temperature of a blackbody at the top of the moon's atmosphere.

We determine the habitability of the hypothetical exomoons described in Section 3 by utilizing the methodology applied by Kane & Gelino (2012). They first determined the inner and outer edges of the habitable zone based on the runaway greenhouse and maximum greenhouse effects (Underwood et al. 2003). Then, recognizing the limited amount of information regarding the atmosphere and surface of the exoplanet (in this case exomoon), they calculated a possible range for the equilibrium temperature based on the heat redistribution of the atmosphere. For example, if the atmosphere is assumed to be 100% efficient, the equilibrium temperature is defined as

$$\begin{equation} T_{100\%} = \left (\frac{L_{\star }\,(1-\alpha _p)}{16\,\pi \,\sigma \,r^2} \right)^\frac{1}{4}, \end{equation} \tag{ 3 }$$

where r is the distance between the star and the planet–exomoon system. In comparison, if the atmosphere is not at all efficient in redistributing the heat, such that there is a hot day-side, then $T_{0\%} = T_{100\%} \times 2^{-1/4}$. See Kane & Gelino (2012) for more discussion.

In Table 2 we give the thermal equilibrium temperature range corresponding to both T_100% and T_0% at the inner and outer edges of the habitable zone using Kopparapu et al. (2013) for all four of our application systems. We note that the habitable zone regions for smaller-size planets and moons are slightly different than those calculated here, although not significantly. The equilibrium temperature was calculated from the stellar flux, such that T_eq = (F_s/σ)^1/4, which by far dominates the sources of flux received on the moon's surface.

We have included the average, minimum, and maximum flux and thermal temperatures for the hypothetical moon in each system within the table, as a comparison to the habitable zone equilibrium temperature boundaries. The average equilibrium temperature of the moon around HD 28185 b is warmer than the inner habitable zone boundaries defined by a 100% efficient atmospheric heat redistribution. In comparison, the moons orbiting BD +14 4559 b and HD 73534 b are too cold for a 0% efficiency rate, but have an average thermal temperature within the range for 100% redistribution efficiency. The hypothetical moon surrounding μ Ara b is the only moon with an average equilibrium temperature that falls within the inner and outer habitable zone boundaries for both extremes in the atmospheric heat redistribution.

The differentiation in average equilibrium temperature between the four systems is most likely due to the fractional time that HD 73534 b spends within the habitable zone which, according to Kane & Gelino (2012), is ∼62.7% of its orbital phase. In addition, the moon's semi-major axis is +1.4 times the distance from the planet as compared to the other systems. The large distances from both the star and the planet significantly reduce the radiation experienced on the moon's surface. However, if the hypothetical moon has an atmosphere that is relatively proficient at distributing the heat, then the moon's average thermal equilibrium temperatures are habitable, even if this host-planet isn't within the habitable zone 100% of the time. For all the cases we've examined, the hypothetical exomoon at the tidal locking radius around HD 73534 b is the only exomoon whose equilibrium temperature range lies fully within any of the habitable zone boundaries, in this case, for a fully efficient heat redistribution.

As was discussed in Section 2.3, large planetary eccentricities give rise to substantial ranges in the equilibrium temperature. The extreme thermal temperatures experienced on the moon around BD +14 4559 b, where e_sp = 0.29, make it uninhabitable no matter the heat distribution efficiency and despite the average thermal temperature. For a fully efficient heat redistribution, the thermal temperature of the moon reaches a maximum temperature that is ∼8 K above the inner habitable zone limit and a minimum thermal temperature ∼1 K below the outer habitable zone limit. In this scenario, the increased flux received by the exomoon at the time of the planet's periastron was not counterbalanced by the fact that the planet only spends ∼68.5% of its orbital phase within the habitable zone (Kane & Gelino 2012). As a result, the exomoon experiences an equilibrium temperature swing that spans the entire range of the inner and outer habitable zone limits for a fully efficient heat redistribution. When the atmosphere is 0% efficient, the exomoon is in general too cold to be habitable, such that the maximum temperature lies nearly exactly halfway between the inner and outer habitable zone boundaries (see Table 2).

We find that for HD 28185, the maximum equilibrium temperatures for the hypothetical moon are above, ∼13 K, the inner habitable zone boundaries at 100% efficiency. However, the thermal temperature range is fully encompassed in the inner and outer habitable zone boundaries for 0% efficiency. In contrast, the moons around BD +14 4559 b and HD 73534 b have equilibrium temperatures more closely aligned with an atmosphere that is completely efficient at redistributing heat. The thermal temperature ranges experienced by both moons are almost fully within the inner and outer habitable zone boundaries at 100% heat redistribution. It is only the exomoon orbiting around μ Ara b that is too hot when there is fully efficient heat redistribution (maximum thermal temperature 5 K above the inner habitable zone) and also too cold when there is no heat redistribution (minimum thermal temperature 9 K below the outer habitable zone). In other words, the thermal equilibrium temperatures would fall within the inner and outer habitable zone boundary for a heat redistribution between 0% and 100%.

We have analyzed four physical systems with hypothetical moons near the Hill or tidal locking radius. By examining the thermal equilibrium temperature range imposed by the geometry of the system, namely the minimum, maximum, and average, we find that the equilibrium temperature of the moons fall within habitable limitations during two scenarios. The first scenario occurs when the host-planet is fully within the habitable zone and the heat redistribution on the moon is relatively inefficient (<50%). Second, the moon is habitable when the host-planet spends only a fraction of its phase in the habitable zone and the heat redistribution on the moon is closer to 100% efficiency. In those cases where the equilibrium temperature extrema brought on by high planetary eccentricity cannot be subdued by less time (or orbital phase) in the habitable zone, the exomoons may not be habitable. By looking specifically at exomoons at the furthest stable/locked radii from the host-planet, we have placed an upper bound on the range of expected thermal equilibrium temperature profiles for rocky moons.

The thermal equilibrium temperature of an exomoon is affected by the both the host-star and -planet. However, there are a number of extreme scenarios that would maximize the received flux on the moon and consequently increase its equilibrium temperature toward habitability. If we were to change the parameters of the star–planet–moon system, then an increased stellar luminosity or planetary radius would bolster the thermal temperature of the exomoon. Decreasing the distance between the planet–moon system and the star, or even the distance between the planet and the moon, would also have a similar effect. In addition, if we allowed the exomoon's eccentricity to become non-zero, then there would be an increase in tidal heating between the planet and the moon (Heller & Barnes 2013).

However, given the confines of our study which has been to analyze physical, stable star–planet systems where the hypothetical moons are located relatively far from the host-planet, then we are left with few ways in which to maximize the equilibrium temperature of the exomoon from contributions by the host-planet. If the planet was relatively young and enriched with radioactive isotopes (such as ²⁶Al or ⁶⁰Fe), then the temperature of the planet would be greatly elevated to the point where volatiles were evaporated from the surface (Grimm & McSween 1993). The exomoon would experience this thermal planetary increase to the power of four via the flux; however, the short-lived nature of the isotopes would mean that the planet and exomoon would cool after ∼1 Myr. A giant impact colliding with the planet may also have a similar effect, i.e., causing the planet's temperature to drastically rise, perhaps through the release of magma, but then eventually cool with time. Another scenario depends on whether the planet is tidally locked to the star, creating a distinct bright side and dark side. The thermal flux received by the exomoon would be increased if the orbital angle of the exomoon was adjusted in such a way as to maximize the contribution of the bright side's significantly warmer (as opposed to the dark side's) thermal irradiation onto the exomoon. Finally, changing the albedo of the planet would alter both the thermal and reflected planetary flux contribution onto the exomoon, for example by a large impact or on a terrestrial planet via volcanic eruptions or vegetation. Through these more extreme scenarios, the host-planet alone may raise the equilibrium temperature of the exomoon toward habitability, although either for relatively short periods of time or through less significant contributions.