Hovering hummingbird wing aerodynamics during the annual cycle. II. Implications of wing feather moult

Birds usually moult their feathers in a particular sequence which may incur aerodynamic, physiological and behavioural implications. Among birds, hummingbirds are unique species in their sustained hovering flight. Because hummingbirds frequently hover-feed, they must maintain sufficiently high flight capacities even when moulting their flight feathers. A hummingbird wing consists of 10 primary flight feathers whose absence during moult may strongly affect wing performance. Using dynamic similarity rules, we compared time-accurate aerodynamic loads and flow field measurements over several wing geometries that follow the natural feather moult sequence of Calypte anna, a common hummingbird species in western North America. Our results suggest a drop of more than 20% in lift production during the early stages of the moult sequence in which mid-wing flight feathers are moulted. We also found that the wing's ability to generate lift strongly depended on the morphological integrity of the outer primaries and leading-edge. These findings may explain the evolution of wing morphology and moult attributes. Specifically, the high overlap between adjacent wing feathers, especially at the wing tip, and the slow sequential replacement of the wing feathers result in a relatively small reduction in wing surface area during moult with limited aerodynamic implications. We present power and efficiency analyses for hover flight during moult under several plausible scenarios, suggesting that body mass reduction could be a compensatory mechanism that preserves the energetic costs of hover flight.


Introduction
The primary role of flight feathers is to form the surfaces over which lift is generated [1,2]. During the course of the year, these feathers are prone to wear and damage due to a variety of processes, including collisions with vegetation, friction with adjacent feathers and skin and the activity of parasites such as feather lice [3,4]. A possible implication of feather wear concerns its effects on flight performance, as was previously observed in pigeons where feather wear decreased lift production by more than 30% [5]. Similarly, Swaddle et al. [6] suggested that worn plumage may reduce a bird's take-off speed and manoeuvrability, hampering its escape flight performance. Because feather wear is irreversible, feathers must be replaced to ensure their functionality [3,7].
Feather damage is determined by several extrinsic (e.g. habitat type and radiation intensity) and intrinsic (e.g. feather durability) factors. Therefore, bird feather moult properties are shaped by these factors through natural selection which acts on feather-related functions such as flight performance, plumage attractiveness during mating (sexual selection) and body thermal regulation. Feather moult properties, including the sequence, timing, duration and extent of the moult process, may consequently differ in each bird species and even between different populations according to different factors such as body size, life-history properties and diet [3,[7][8][9][10][11].
Feather moult is considered one of the most energy-demanding processes in the annual cycle of birds, and as such, it does not usually overlap with reproduction or migration which constitute other major, annual, energy-demanding processes [7,12]. During moult, reduced thermal insulation from the plumage and the energy cost of feather synthesis have been found to increase bird metabolism by 20-40% above the bird's basal metabolic rate [13]. Furthermore, due to morphological deficits that inevitably result from feather shedding during moult, birds may need to exert more power for flight during the extended periods over which feather replacement takes place [14,15]. The consequences of feather moult on bird aerodynamics are rarely described [7, 14,16], and there is limited evidence regarding metabolic consequences of moult due to the smaller surface area of the wings. While both Epting [15] and Chai [17] found substantial increases in mass-specific metabolic rates of moulting hummingbirds, when considering the whole organism rather than mass-specific metabolism, the latter study does suggest that the flight metabolic rate of moulting hummingbirds is nearly constant. This is achieved through reduction of bird body mass during moult (see below). A number of studies have proposed that feather gaps in the wings and tail that are created during moult reduce lift [16,[18][19][20][21], providing a plausible connection between wing surface deficits, deteriorated wing aerodynamics and elevated flight metabolism of moulting birds.
We studied the consequences of wing feather moult in Anna's hummingbird (Calypte anna). The moult sequence of Anna's hummingbird was studied by Williamson [22] and is described as follows. The first feathers to be shed are primaries (denoted by P; figure 1) 1, 2, 3 and 4, with some variation between different individuals regarding the number of feathers that are shed simultaneously (2-4 feathers). After these feathers are fully regrown, primaries 5, 6, 7 and 8 are replaced one after the other. The secondary feathers (denoted by S; figure 1) are shed simultaneously with primary 6. The 10th primary flight feather is shed after primaries 1 through 8 are fully regrown, and P9 is shed after P10 grows to at least one-fourth of its final size. The disruption of the outwards moult sequence of the primary flight feathers, where P10 is shed before P9, is an exception from common moult sequences in nearly all other bird species [3,22,23]. We note that this moult sequence also characterizes other hummingbird species [24].
Hummingbirds are known to vary their body mass at different times of the day, notably at dusk when birds may be involved in intense feeding that could result in up to a 25% increase of their body mass [25,26]. Also, hummingbird body mass increases when the birds prepare for migration and put on weight in the form of lipids that are later used as fuel during long-distance flight [26,27]. During moult, birds were found to decrease their body mass, thereby alleviating aerodynamic forces required for weight support. Indeed, anecdotal findings, both in hummingbirds [15,17,28] and in other taxa (e.g. [29]), suggest that moulting birds reduce their body mass considerably, possibly indicating a general mechanism of mass support modifications in relation to wing surface morphology. Hummingbirds of several different species, inhabiting a wide breeding range in North America, demonstrated this pattern when examined under controlled conditions in the laboratory [15,17,28]. According to Chai [17], while wingbeat kinematic variables, such as flapping frequency and amplitude, hardly changed throughout the moulting sequence, a 25% reduction in body mass from that in other periods of the year was measured for moulting ruby-throated hummingbirds (Archilochus colubris). Information regarding the extent of this phenomenon during moult among free-ranging hummingbirds and the factors that govern it are still understudied. Moreover, the aerodynamic mechanisms that govern lift production and that may link  Figure 1. An Anna's hummingbird wing consists of 10 functional primary flight feathers denoted by P that are numbered outwards. The secondary flight feathers are denoted by S without numbering specific feathers.
wing geometries of moulting birds to reduced weight support and elevated metabolism have not yet been thoroughly investigated. Therefore, we herein explore the aerodynamic consequences of the natural sequence of flight feather moult of Anna's hummingbird using dynamically scaled hummingbird wings in fluids.

Methods and materials
To study the effects of moult on the aerodynamic characteristics of Anna's hummingbird wings during hovering, time-accurate aerodynamic loads and flow field measurements were made over dynamically scaled-up model wing pairs. The models were suspended in a tank filled with a working fluid, in a propeller-like set-up, where each pair represented a different stage in the moult sequence (figure 2). The wing pairs then followed stages of initial acceleration, steady rotation and deceleration to model the downstroke of a flapping wing. The experimental set-ups have been described in detail by Achache et al. [30] and are briefly described below.

Wing design
Based on three-dimensional laser scans of a complete, dried male Anna's hummingbird wing [31], a 3.5-1 dynamically scaled wing model was fabricated (one wing span R = 175 mm) using stereolithography (Objet VeroBlack TM ). Similarly, following the natural moult cycle of this species [22], three representative moult stages, namely P1-4 (the shedding of primaries 1-4), SP6 (the shedding of the secondaries and primary 6) and P10 (the shedding of primary 10), were modelled by removing the corresponding flight feathers from the three-dimensional, complete wing computer-aided design model. This process was done while taking into account feather overlap (figure 2). Because the secondary flight feathers have been shown to play a minor role in lift production, their individual moult pattern was not examined in this study [17].
The wing structure of Anna's hummingbird is characterized by an increasing overlap of the primary flight feathers from the inner wing outwards, such that the feathers forming the wing tip and leadingedge largely overlap with neighbouring primary feathers (see fig. 1a in [30] or e.g. [32,33]  and damage. The increased overlap of the outer primary feathers may allow the bird to maintain nearly all of its wing surface even when one of the feathers is subjected to substantial damage. Furthermore, the importance of the outer primary flight feathers to flight performance is reflected by the disruption in the outward sequence of moult. All of the above suggest that wing integrity is aerodynamically critical at the leading-edge and wing tip. Therefore, we further studied the role of the wing tip in wing aerodynamics by introducing an additional wing configuration that is not part of the natural moult sequence. This was done by removing the 8th and the 9th primary flight feathers (hereafter P89 wing; figure 2e) in contrast to the natural moult sequence in which only one of these flight feathers is replaced at any given time.
Previous studies suggested that hummingbird wings are rigid throughout the downstroke [32,34,35]. Moreover, it was demonstrated that the porosity of the wings has very little, if any, effect on their aerodynamic performance [31,33]. The above justified the use of a rigid and non-porous, threedimensional printed wing model to analyse Anna's hummingbird's complete wing performance and aerodynamics. To the best of our knowledge, there are no published data regarding these properties (wing deformation and air transmissivity) for a hummingbird wing during active flight feather moult. Therefore, the wing is assumed to be rigid and non-porous throughout the annual cycle for the purpose of presenting a first attempt to understand the consequences of flight feather moult on the wing's aerodynamic performance. It is understood that the results presented herein might be an overestimation of the wing's performance, which needs to be assessed by a designated study.

Force measurements
To measure time-accurate aerodynamic loads acting on the wing during the downstroke, the mechanical assembly was mounted on a six-component force and moment transducer (MINI40, calibration SI-20-1 by ATI Industrial Automation, Apex, NC, USA) that recorded the aerodynamic loads that acted on the wings (see fig. 1 in [30]). To match the Reynolds number of an actual Anna's hummingbird wing, as well as the transducer's effective range of measurements, the wing pair was submerged in a waterglycerin solution (ν = 4.6 cSt). The time-accurate aerodynamic lift, L, and torque, Q, acting on one wing are referenced to 1 2 ρSθ 2 ∞ R 2 2 and 1 2 ρSθ 2 ∞ R 3 3 , respectively. These yield the dimensionless lift and torque due to drag coefficients, C L and C Q , where ρ is the fluid-specific weight andθ ∞ is the wing's rotational rate at mid-stroke. S, R 2 and R 3 are the area and the radii of the second and third moments of area of the complete wing, respectively. The wings were tested at various angles in relation to the wing root (α), ranging from 5 • to 35 • at 5 • increments. For a better signal-to-noise ratio, each experiment was repeated 50 times (n = 50). The time-averaged aerodynamic coefficient for each individual load measurement i where * = {L, Q}, T is the downstroke duration (T = 0.74 s) and τ is the integration variable. Timeaveraged aerodynamic coefficients are subsequently defined as At the end of each wing stroke, a pause of approximately 30 s took place to establish an essentially quiescent flow at the beginning of each wing stroke. The time-averaged lift coefficient required to support the bird's body weight (divided by two, in order to refer to one wing) isC where m b is Anna's hummingbird's body mass during the moulting period, g is the gravitational acceleration, ρ a is the air density at sea level and f b is the wingbeat frequency, defined as 40 Hz [36]. S b and R 2 b are the complete wing area and radius of the second moment of area, respectively [30].

Flow field measurements
Flow field measurements were obtained at α = 30 • using phase-locked particle image velocimetry (PIV). Our experimental set-up consisted of a PIV system (Dantec Dynamics A/S, Skovlunde, Denmark) comprising a dual-cavity 30 mJ Nd:YLF laser with a repetition rate of up to 10 Hz, a 4M pixel resolution 12 bit CCD camera (FlowSense EO) and a programmable timing unit. Two experimental schemes were applied. The first measured the in-plane velocity component at different angular positions along the downstroke (θ ) over three span stations, namely, z = r/R = 0.25, 0.5 and 0.75, where r is the dimensional distance of the span station from the wing's root. The second measured the near-wake velocity field at representative downstroke stations (see fig. 2 in [30]). The dimensional unsteady velocity field is defined as where U i is the flow field that was acquired at the ith measurement (1 ≤ i ≤ n). U, V and W are the velocity components in the X, Y and Z directions, respectively (see fig. 1 in [30]). The tangential velocity at each span station is U (z) =θR(z + η), where η is the dimensionless distance between the wing's root and the centre of rotation (referenced to R; see fig. 1d in [30]). The dimensionless velocity field in the wing's frame of reference is defined as u = {U − U , V, W}/U = {u, v, w}. The near-wake velocity fields refer to the wing tip velocity, u * = u/U tip , where U tip =θR(1 + η). For statistical analysis, the velocity fluctuations are defined by using the standard definition referring to the wing tip velocity at mid-stroke, We define a standard deviation operator of the two components of the vector field as σ u 1 u 2 = [ 1 2 (u 2 1 + u 2 2 )] 1/2 .

Aerodynamic loads
The time-averaged aerodynamic coefficients during the downstroke are presented for all tested moult stages in figure 3a. As expected, the complete wing produced the highest lift when compared with the wings of the three tested moult stages. At the first stage of the moult sequence, when the four innermost primary wing feathers (P1-4; figure 2b) have been shed, 22% of the wing surface is missing (table 1), constituting the largest area loss during the entire sequence. Lift production, at this stage, declined by 0.4Ĉ L (hat denotes unit value), compared to the complete wing for α > 10 • . In the next moult stage, when P6 and the secondary flight feathers are missing (SP6; figure 2c), a 10% area loss at the root and middle of the wing induced a reduction in lift capacity similar to the previous moult stage. Bridge [5] measured similar magnitudes of lift reduction over a model of a pigeon wing with gaps in the mid-wing primaries.
After P1-8 have fully grown in, the ascending order in which flight feathers are replaced is reversed, and P10 is shed. The rachis of P10 extends from the wing's medial section (z ∼ 0.4) to the tip, and its vane forms the sharp leading-edge of the wing. Therefore, when shed, along with a 0.87% wing surface reduction (table 1 and figure 2d), the leading-edge thickness and shape are also altered. Because, to the best of our knowledge, the leading-edge thickness of an Anna's hummingbird wing was not measured during active moult of the P10 flight feather, in our study the thickness is a resultant of the manufacturing process as described earlier (complete, approx. 0.5 mm; P10, approx. 2 mm). In a previous study, conducted over revolving hummingbird planform wings mounted at α = 45 • , lift production was   not shown to be sensitive to the change in leading-edge thickness [38]. However, over the range of tested angles of attack in this study, a reduction of 0.11Ĉ L is evident for 15 In revolving wings, the tangential velocity grows linearly along the span. Therefore, loss of the medial and outer primaries would probably impair lift production more than the absence of primaries closer to the root. The moult sequence seems to follow this logic as the proximal feathers (i.e. P1-4) are shed simultaneously, opening the largest gap in the wing during the moult cycle. This is followed by a sequential replacement of the medial and outer wing feathers, where smaller gaps are opened due to both fewer feathers being shed simultaneously and the increasing overlap between adjacent feathers. To assess the effects of gap size and position in the wing on lift production, the lift coefficient for each moult stage was also calculated while taking into account each stage's corresponding wing geometry. This was done as follows:C  (as low as 50% compared with the complete wing; figure 3b). The degraded performance of this moult stage is associated with a large gap opened in the wing surface with respect to the effective disc area (see fig. 5 in [30]). As most lift production is made by the medial wing section, gaps in this area cause a substantial reduction in the ability of the wing to generate lift. At a later stage of the moult sequence, namely P10, the missing surface area at the distal part of the wing (where the tangential velocity is the highest) was smaller overall, such that the wing maintained its ability to produce lift. Owing to the increasing overlap and the gradual sequence in which the medial and outer flight feathers are replaced, it is likely that if larger gaps were naturally found in the hummingbird wing as part of the regular moult sequence (e.g. through simultaneous shedding of two or three adjacent feathers), lift production would have been substantially impaired. This aspect is treated below through the introduction of the P89 wing configuration ( figure 6). Throughout the moult stages, torque values have proportionally decreased relative to the reduction in lift described above, indicating a link between lift and torque (figure 3a). Hummingbird wings have been shown to have superior lift-to-torque ratios over wings consisting of an attached leading-edge vortex as their governing aerodynamic mechanism (i.e. insects [39][40][41][42][43][44]) due to the development of a leading-edge suction force (C L /C Q > cot α [30,37,45]). Here, the P10 wing showed similar values of lift-to-torque ratios throughout the spectrum of angles of attack as that of the complete wing, outperforming the geometrical prediction for α > 10 • ( figure 3c). Surprisingly, despite the substantial gaps in the P1-4 and SP6 wings, for α > 15 • , these wings also outperformed wings consisting of an attached leading-edge vortex. The higher angle of attack at which they succeeded in doing so, relative to the complete wing, serves as an indication of the effects of the large gaps in delaying the formation of a leading-edge vorticity structure. For α > 20 • , lift-to-torque ratios converged at the different moult stages and are analogous to cot α, suggesting the presence of a leading-edge bubble over the wing throughout the moult sequence (figure 3c) [30].  different moult stages and the wing aerodynamic characteristics. In light of the large reduction in lift capacity at the early stages of the moult cycle, when wing area is the smallest, we present near-wake measurements over a wing with missing P1-4 in figure 4. These measurements present the development of the tip vortex throughout the downstroke along with the effective disc area created over rotating or flapping wings and are characterized by negative vertical velocity values associated with lift. To enable a comparison to the complete wing, the effective disc area is provided in relation to the wing

Flow field measurements
Apart from the well-established primary tip vortex which formed at the wing tip at the early stages of the downstroke, a smaller, clockwise, secondary vortex appeared in the middle of the gap, effectively acting as a tip (θ = 22.5 • , figure 4a,b). The effective disc area is enclosed between the two vortices (primary and secondary) over the medial part of the wing, covering 65% of the wing disc area (figure 4a). At midstroke, the tip vortex is fully detached and fluctuating flow progressed from tip to root ( figure 4c,d). The effective disc area migrated up the span and eventually covered 42% of the wing disc area. Overall, the missing surface of the wing induced a 16% reduction in the effective disc area compared to the complete wing throughout the downstroke ( fig. 5 in [30]). The reduction in effective disc area yielded a qualitative measure of the lift degradation as evident by the aerodynamic loads obtained over the P1-4 wing (figure 3a).
To further understand the effect of moult on the main flow features, a comparison was made between the flow fields measured at mid-span and mid-stroke (z = 0. We examined the significance of leading-edge and wing tip integrity by measuring the near-wake flow field of the P89 wing ( figure 6). In this wing configuration, the wing is missing 11% of its surface area (table 1), and the gap is mostly found over the distal section of the wing (figure 2e). We found that the gap in the wing caused the tip vortex to break down even in the early stages of the downstroke ( figure 6a,b), unlike the well-structured tip vortex that was present over the complete [30] and the P1-4 ( figure 4a,b) wings. As a result, the effective disc area, stretching between the tip and root vortices, covered 0.63S d , a 19% reduction compared to the complete wing at this stage. Further down the stroke ( figure 6c,d), the effective disc area covered an area of 0.2S d , a 40% decrease compared to that of the complete wing at this stage of the downstroke. As the effective disc area is associated with lift production, the large reduction in its size indicates a substantial aerodynamic impairment for this wing configuration. The distribution of the vertical velocity along the span of the P89 and complete wings further demonstrates the reduction in lift when the integrity of the wing tip is impaired (figure 6a,c). Although similar values of negative vertical velocity were measured within the effective disc area, beyond it, weaker negative

Flow field analysis
To investigate the effect of feather moult on the flow mechanisms over the Calypte anna's wing and, subsequently, its connection to the wing's aerodynamic characteristics, below we present the spanwise vorticity field over the wing and the circulation at several span stations. A similar analysis was performed for the complete wing ( fig. 6 in [30]).

Late stages of moult
During the later stages of the moult sequence, represented by the P10 moult stage, the flight feathers forming the tip and the leading-edge of the wing are replaced. As suggested by the flow fields over mid-span at mid-stroke (figure 5c), a large leading-edge bubble developed over the P10 wing's leadingedge, covering about 0.6c and was maintained throughout the remainder of the downstroke (figure 8a-c). Interestingly, despite the seemingly large differences in flow field development (figure 5), the circulation measured over the P10 wing is almost identical to that which was measured over the complete wing throughout the entire downstroke (figure 8g), as was also suggested by the lift force measurements ( figure 3a). Yet, at 22.5 • < θ < 135 • , the circulation measured over the P10 wing is lower than that of the complete wing, showing an 8% decrease in circulation at mid-stroke. This finding agrees with the force measurements that were recorded for the P10 wing ( figure 3a).
Regarding the flow field at the distal span station, the vorticity field was measured to be essentially the same as that of the complete wing as the flow fully detaches along the downstroke (figure 8). The circulation reaches its maximal value at this station just after the end of the acceleration stage (figure 8g). This is followed by an abrupt drop in circulation associated with flow detachment over the tip with the same magnitude as that recorded over the complete wing. Throughout the remainder of the downstroke, the overall circulation over the distal span station is as measured over the complete wing. production as indicated by the reduced effective disc area ( figure 6). These results are in agreement with Chai [17], who reported the effect of gravely reduced hovering aerodynamic capacity when the wing tip of the ruby-throated hummingbird is cut off. Despite the ability to withstand larger morphological deficits to the wings during moult, a wing area loss of 10% at the wing tip resulted in failure to sustain hovering flight [17].

Wing performance during moult
The downstroke aerodynamic characteristics of Anna's hummingbird wing during moult of its primary flight feathers are presented in figure 9. We distinguish two underlying factors to estimate the power requirements while hovering with a moulting wing, the bird's body mass (m b ) and the downstroke For the standard deviation of the complete wing and wing kinematics, see [30].
contribution to the overall lift (and torque due to drag) produced by a wingbeat (downstroke and upstroke), h [30]. While numerous studies have discussed and evaluated these properties for hummingbirds during non-moulting periods of the year (m b = 4.6 g [47]; 0.66 ≤ h ≤ 0.75 [48][49][50][51]), there is neither a comprehensive nor reliable database for these parameters during moult. Regarding body mass, Chai [17] has reported that ruby-throated hummingbirds decrease their body mass by about 25% during moult at stages associated with wing area loss corresponding to the P1-4 and SP6 wings in our study. Similar results, albeit of a somewhat lesser degree of mass loss, were reported for rufous hummingbirds (Selasphorus rufus [28]). Epting [15] reported a 10% decrease in body mass of an individual black-chinned hummingbird (Archilochus alexandri) after breaking the distal tip of the 10th primary flight feather and different extents of body mass reduction at different moult stages The shaded red region represents the lower downstroke specific power muscle limitations adopted from [46]. Grey areas are the acceleration and deceleration phases of the wing motion. For wing kinematics, see [30].
on an individual Allen's hummingbird (Selasphorus sasin). However, as no equivalent data exist for Anna's hummingbirds, it is unknown if this species decreases its body mass during moult and to what extent. Furthermore, it remains to be seen if the birds reduce their body mass during moult regardless of their moult stage, or if they adjust their body mass in correspondence with the wing surface area during different stages of the moult sequence as was suggested by Epting [15]. If the latter is true, aerodynamic forces required for weight support may vary considerably during the entire moulting period because the hummingbird moult cycle involves reduction of wing area to different extents [22]. Moreover, hummingbird body mass varies significantly throughout the day, with fluctuations of up to 25% [25,26]. These uncertainties led us to suggest three scenarios which confine our aerodynamic performance analysis: (1) no body mass reduction during moult (m b = 4.6 g [47]); (2) a 25% body mass reduction, corresponding to Chai's findings [17], which is assumed to be constant throughout the entire moulting period; and (3) a scenario similar to Scenario 2 but with a 10% body mass reduction at the P10 moult stage as inferred from Epting [15]. As presented in figure 3, the wing's capacity to generate lift (i.e. weight support) during the downstroke is reduced during moult. It is unknown, though, if the degradation in lift produced by  the upstroke is proportional to that of the downstroke. For example, slits opened in the wing during the upstroke (between the S1 and P1 flight feathers [32]) might grow larger during moult, resulting in wing deformation and area reduction larger than the sum of the missing feathers, consequently affecting the lift produced during this phase. If so, h might change between periods in which the wing is complete and during moult (h will increase when the downstroke must account for more weight support). Furthermore, it is possible that each moult stage has its own value of h. To represent the variability of h, the aerodynamic performance estimation is presented for 0.66 ≤ h ≤ 0.75 in figure 9 and table 2 based on the range reported for the complete wing [48][49][50][51].
To provide sufficient lift during the downstroke,C L = 2hC L b [30]. In the case of no body mass reduction during flight feather moult (Scenario 1), a mean value ofC L = 1.4 ± 0.1 (mean ± s.d due to h variation) is required from the downstroke in order to maintain hovering flight (figure 9a). At such conditions, Anna's hummingbird wing lift-to-torque ratio (figure 9b) and its ability to support a unit of weight (PF [30], figure 9c) decreased by about 33% and 50%, respectively, compared to that of the complete wing. At a moult stage where the P10 feather is missing, the wing performed similarly to the complete wing (figure 9a-c). Assuming that the bird reduces its body mass by 25% during the entire moult period (m b = 3.45 g, Scenario 2), a more moderate downstroke average lift coefficient is required to sustain hovering flight,C L = 1.1 ± 0.1 (figure 9a). This allowed the wing to perform at a more efficient working point, as can be inferred by the higher values of the lift-to-torque ratio and PF (figure 9b,c).
One of the dimensional parameters that may provide useful insights regarding Anna's hummingbird flight performance is the aerodynamic power required to sustain hovering flight. The downstroke timeaccurate muscle mass-specific power (P * ; see [30] for the mathematical expression) for Scenarios 1 and 2 at the tested range of h (at the corresponding downstroke average angles of attack, α; figure 9a) is presented in figure 9d-g. Previously reported flight muscles averaged power limitation for different species of hummingbirds [46] were modified to correspond to the pectoralis (downstroke flight muscles) specific power limit (see eqn. 4.6 in [30]). Doing so allowed us to offer a benchmark value for the power requirements estimated during moult for the different scenarios. Downstroke time-averaged specific power, defined asP * = T 0 P * /T, is provided in table 2 for each scenario. Following Scenario 1 and h = 0.75, the relatively low efficiency induced a high aerodynamic power requirement for the P1-4 and SP6 wings-almost two times higher than that of the complete wing (table 2) and greater than the downstroke flight muscle power limit throughout most of the downstroke (figure 9d). On average, flapping at h = 0.66 (figure 9e) decreased the cost of hovering by about 10% compared with that of the higher h factor, within a safety margin from the power limitation. Reducing body mass by 25% (Scenario 2) decreased the required power below that of the complete wing ( figure 9f,g). Furthermore, the wing performed closer to its optimal PF, making the specific power requirement less sensitive to variations in h (figure 9c, table 2). We note that the estimated wingbeatspecific power requirements (P * wb ; see [30] for the mathematical formulation) in this scenario for the P1-4 and SP6 wings are 69.4 ± 6.7 W kg −1 and 70.7 ± 4.7 W kg −1 , respectively. These values are in very good agreement with Chai's estimation of moulting ruby-throated hummingbirds, with similar wing areas, assuming flight muscles mass is fixed during moult (72 ± 6 W kg −1 and 69 ± 3 W kg −1 [17]). These results support our experimental approach and provide a reliable estimation of the entire wingbeat performance during moult.
Interestingly, throughout Scenarios 1 and 2, the power requirements during the P10 moult stage are lower than the flight muscles limit. Reducing body mass according to Scenario 2 even reduced the specific power requirements by more than 33% compared to that of the complete wing ( figure 9f,g and table 2). Therefore, mass reduction at the P10 moult stage is not a necessity in terms of aerodynamic performance in hovering. Applying a 10% body mass reduction at the P10 moult stage (Scenario 3) yielded power requirements which are similar to those of the complete wing and the rest of the moulting stages with a 25% body mass reduction (table 2). Under these circumstances, flight costs throughout the year vary by several per cent, as a result of properly adjusted body mass. Thus, one may infer that the extent of mass reduction during moult of the primary flight feathers is not constant and is determined by the basal (complete wing) flight cost.

Conclusion
Our findings provide inferences regarding the effect of moult on hummingbird wing aerodynamics. Our results suggest that depending on the moult stage, lift production can drop by more than 20% compared to the complete (non-moulting) wing. The largest effect of feather moult on wing performance was recorded when the medial flight feathers were missing, inducing large flow fluctuations that reduced the extent of the effective disc area. The feathers constituting the medial section of the wing, namely P5-7, are sequentially shed only after P1-4 are fully regrown [22], presumably as an outcome of natural selection to avoid gaps that may induce a substantial reduction in wing performance to an extent that may even cause aerodynamic failure.
The outer primary flight feathers are shed at the end of the moult sequence after the rest of the wing is renewed. Although the missing outer primary feathers at the later stages of the moult did not significantly affect lift production, the wing tip feathers were found to be essential for lift production. Flow field measurements over a wing with two outer primaries missing (P89) indicated a substantial drop of up to 40% in the effective disc area compared to that of the complete wing. This suggests that Anna's hummingbird wing performance is highly sensitive to morphological deficits at the wing tip. The importance of the integrity of the wing tip explains the increasing overlap between adjacent flight feathers towards the tip of the wing and the alteration of the moult sequence in Anna's hummingbirds [22], as well as in other hummingbird species [24], in which the outward moult sequence changes when reaching P9, minimizing the moult gap area at the wing tip.
Power and efficiency analyses are provided for a range of scenarios in which hummingbirds may operate during flight feather moult. These analyses suggest that Anna's hummingbirds may need to reduce their body mass during moult, at least at the early stages of the moult sequence, to withstand the limitations imposed by their flight muscles that must operate with reduced effective disc area. Furthermore, body mass variation may be governed by the interplay between biomechanical, physiological and behavioural factors and not only by power requirements during different periods of the year and even between different times of the day. To this end, we encourage further research on body mass variation and concomitant moult extent during the moulting period of free-ranging hummingbirds to enhance our understanding of the required weight support and aerodynamic and metabolic consequences of body mass and wing morphology at multiple time scales.