Whiteflies stabilize their take-off with closed wings

When flying animals take off, the lift produced needs to overcome the weight of the flyer and accelerate the body upwards. The maximal upwards acceleration should be limited by the aerodynamic performance of the wings and the maximal power output of the flight muscles (Marden, 1987). However, pushing against the ground with the legs can be used to achieve steeper and/or faster take-offs (Bimbard et al., 2013; Pond, 1972; Provini et al., 2012). The contribution of the leg thrust for take-off may range from 0% in animals that rely solely on the wings (e.g. rhinoceros beetles; Van Truong et al., 2014), to 59% in hummingbirds (Tobalske et al., 2004), which use both legs and wings, and up to 100% in animals that launch themselves by jumping into the air with the wings folded next to the body (e.g. grasshoppers; Pond, 1972).

Flight initiation by jumping is particularly interesting in insects, because within the animal kingdom, arthropods hold the record in jump performance relative to their small body size (Schmidt-Nielsen, 1997). Insects such as fleas (Bennet-Clark and Lucey, 1967), grasshoppers (Bennet-Clark, 1975; Heitler, 1974), leafhoppers (Burrows, 2007), flea beetles (Furth et al., 1983), click beetles (Evans, 1972) and many more (see review in Burrows and Dorosenko, 2014) can launch their body into the air, covering distances of tens of body lengths in a single jump. To increase the speed at leaving the ground, some insects push their body off the ground with extremely high acceleration, reaching up to 300–400 times the gravitational acceleration (e.g. click beetles and froghoppers; Burrows, 2003; Evans, 1972). To provide the high power needed for these accelerations, these insects rely on elastic energy (Gronenberg, 1996; Patek et al., 2011). In such jumps, muscles contract slowly to perform mechanical work on an elastic element of the skeleton. The elastic element serves as a spring, storing the mechanical work, until a trigger mechanism abruptly unleashes the stored elastic energy, launching the insect into the air at striking accelerations (reviewed by Gronenberg, 1996).

If the abrupt ground reaction force, which is generated during the jump, does not pass exactly through the center of mass, a high torque will ensue, sending the insect tumbling into the air. Indeed, vigorous rotations (10–400 revolutions s⁻¹) in the air have been described in jumping insects that propel their jump elastically (e.g. flea beetles; Brackenbury and Wang, 1995; springtails; Burrows, 2012; jumping plant lice; Christian, 1978). Because it is difficult to avoid the torque generated during take-off, insects that jump to initiate flight should be able to correct body orientation during the transition from jumping to stable flapping flight (Brackenbury and Wang, 1995; Card and Dickinson, 2008). With their wings spread out, insects can use their wings to dynamically correct for perturbations in their body orientation (Dickinson, 1999; Ristroph et al., 2010, 2013), enabling them to aerodynamically stabilize the orientation of their body. However, when the take-off is performed with the wings placed in the resting position, the ability to control body orientation in air may be limited.

We investigated how whiteflies cope with this problem during their take-off jumps with closed wings. Whiteflies (Hemiptera: Aleyrodidae) are notorious agricultural pests, and their biology and flight behavior have been studied extensively (Blackmer and Byrne, 1993; Byrne, 1991; Isaacs et al., 1999). However, with the exception of one qualitative description (Weber, 1931, see Discussion), very little work has been done on their flight initiation mechanism. While working on the take-off of the tobacco whitefly, Bemisia tabaci (Gannadius 1889) (adult body length ca. 1 mm), we noticed that they launch their body into the air by jumping with their wings held against the sides of the body in the resting position. The wings are elevated to the dorsal position, in preparation for the first downstroke, only after the body had moved several body lengths through the air (Movie 1). The jumps are abrupt and powerful (see Results), suggesting that whiteflies exploit elastic energy for jumping.

We hypothesized that the powerful take-off jumps pose a challenge to stabilizing body orientation, and that the long wings, in their resting position, act in the air to stabilize body orientation prior to wing deployment and the initiation of flapping flight. To test this hypothesis, we recorded whitefly take-off using high-speed cameras. We then analyzed the resulting movies to time wing deployment and body rotation during the transition from the ground to flapping flight. We expected to find in each take-off incident one of the following scenarios. (1) Whitefly transition from ground into air without substantial rotation of the body, indicating that whiteflies push against the ground in a stable manner that prevents body rotation during the transition from ground into air. (2) Whiteflies tumble in the air immediately after the take-off jumps, stopping the rotation of the body only after the wings are deployed and flapped. This would indicate that a destabilizing torque is generated as the insect pushes against the ground, and that the insect relies on the aerodynamics of the flapping wings to stabilize body rotation. (3) Whiteflies leave the ground tumbling but stop or slow down the body rotation considerably prior to wing deployment and flapping. This would suggest that destabilizing torques are generated during pushing against the ground, but also that once the body is in the air, aerodynamic counter-torques, generated from the motion of the body through air, stabilize body rotation prior to wing deployment. Such counter torques can result from longitudinal static-stability where aerodynamic forces acting on the body and wings at a distance from the center of mass result in aerodynamic counter-torque when the insect rotates (pitch) from a stable orientation (Ribak et al., 2013; Thomas and Taylor, 2001). To evaluate the contribution of the wings (in the resting position) to stabilization of body orientation in the air, we observed the rotational dynamics at take-off of whiteflies with clipped wings.

Tobacco whiteflies, Bemisia tabaci biotype MEAM1 (Fig. 1), were collected using an aspirator from a population maintained at Tel Aviv University in a greenhouse on collard (Brassica oleracea acephala) plants. The insects were transferred to glass vials and transported to the laboratory for testing within the same day. Morphological data were obtained from measurements taken from photographs of cooled whitefly adults, using a camera mounted on a dissecting microscope (Wild M40, magnification ×50). Body dimensions were measured in the images. Forewing length, area and center of area were calculated from the wing's contour in the images, as in Ribak et al. (2009).

To capture the rapid take-offs, we used two high-speed cameras (Fastcam SA3, Photron Inc.) positioned orthogonally, filming at 2000 or 3000 frames s⁻¹. An additional filming session at 6000 frames s⁻¹ focused on the action of the insects on the ground (see below). To provide light for high-speed photography, two near-infrared LED arrays were pointed at the cameras, providing backlight illumination to the filmed whiteflies. The glass vial containing the insects was placed between the lights and the cameras, and was covered by a flat glass lid, leaving only a small opening for the whiteflies to exit the vial. When a whitefly from inside the vial found the small opening in the lid and passed through it, it typically walked on the upper side of the lid for a short period, then paused and took-off. In total, we recorded take-offs from 148 whiteflies.

First, we observed the take-off of 100 whiteflies. Most (91%, see Results) of the whiteflies took off in a jump with their wings in the resting position. The transition from ground to flapping flight followed a pattern consisting of several components in a sequence (referred to as ‘acceleration’, ‘ballistic’, ‘supination’ and ‘elevation’ phases in Fig. 2). First, the insect paused motionless on the ground. Then, in the ‘acceleration’ phase, the center of mass accelerated to take-off velocity (velocity at leaving the ground) as the hind legs pushed against the ground. Next, in the ‘ballistic’ phase, the body moved through the air with the wings in their resting position. In the next phase, wing deployment started with a supination of the wings, while the wings are still pointing backwards. Supination continued until the plane of the wings became parallel to the frontal (dorsal) plane of the body (Fig. 2A). Finally, in the ‘elevation’ phase, the wings were elevated about their joints to the dorsal stroke reversal point without moving sideways. Once the wings reached that position, they pronated and moved down in the first downstroke (supplement film 1). Time allocation between these phases was measured by finding the following time points in the take-off sequence (Fig. 2): (t₀) the first movement of the body during the acceleration phase; (t₁) the instant the last leg left the ground; (t₂) the instant the wings started supinating from their position at rest (along the sides of the body); (t₃) the instant at which the wings in the frontal plane started to elevate; and (t₄) the instant at which the wings reached the final dorsal position and started to pronate in preparation for flapping. Time points t₀ and t₁ defined the duration of the ‘acceleration’ phase; t₁ and t₂ defined the duration of the ‘ballistic’ phase; and t₂ and t₄ defined the duration of ‘wing deployment’, which could be sub-divided to: ‘supination’ (t₂−t₃) and ‘elevation’ (t₃−t₄). The acceleration phase proved to be very short (1.7 ms, see Results). Therefore, to confirm its duration with improved temporal resolution, we filmed the take-offs of another 14 whiteflies at 6000 frames s⁻¹.

where x, y and z denote the coordinates of T and H on the lab-based XYZ axes, where Z is vertical and positive when pointing up. Image resolution in the movies sets a limit to the accuracy of the measurement of θ. The minimal measurement error in each of the movies was on average 2.8 deg (s.d.=0.76 deg, n=34).

where A′ is a non- dimensional number between 0 and 1. A value of A′=1 indicates that the entire wing area is in a plane perpendicular to the direction of motion.

To empirically test the significance of the closed wings in aerial righting, we filmed additional whiteflies taking off, either after their wings were entirely removed or after removing the distal half of the wing length. To cut the wings, the insects were immobilized by cooling to 4°C. Because of their small size and delicate bodies, it proved extremely difficult to cut the wings of whiteflies without injuring them. Furthermore, the uninjured whiteflies were reluctant to jump after their wings were clipped. As a result, we were only able to record the jumps of 34 uninjured insects and each cooperating insect was allowed to take off more than once. This resulted in 13 usable take-offs by seven whiteflies with wings removed and 10 usable take-offs by nine whiteflies with clipped distal wing sections. In movies showing insects with clipped wings, we identified all the take-off phases described above. In movies showing wingless whiteflies, we only differentiated between the ‘acceleration’ and ‘ballistic’ phases, with the latter continuing until the insect left the field of view of both cameras.

To evaluate whether aerodynamic force on the closed wings and body can resist the (forward) pitching rotation of the body in the air, we used a simplified biomechanical model based on the quasi-steady, blade-element approach. The model deals only with the aerial phase of the take-off jump, i.e. the period after the insects leave the ground and prior to the initiation of wing flapping.

where I_yy is the moment of inertia for rotation about the pitch axis passing through the center of mass, τ_w is the aerodynamic torque of the closed wings, τ_b is the aerodynamic torque on the body moving with its longitudinal axis at an angle to the direction of motion, C_s is the rotational damping coefficient of the body and and are the angular velocity and angular acceleration for the pitch rotation, respectively.

where m is the total mass of the body.

where μ is the dynamic viscosity of air (1.84×10⁻⁵ kg m⁻¹ s⁻¹).

where ρ is the density of air (1.2 kg m⁻³), U is the velocity of the body, d_p is the equivalent diameter for a spherical body with the same volume as the ellipsoid and C_p is the torque coefficient.

where w is the transverse axis (diameter).

To estimate the aerodynamic torque of the wings in the resting position (τ_w), we used a blade-element approach. We divided the length of the forewing (the hindwings are tucked under the forewings while in the resting position) into four sections (wing elements, Fig. 1C) and calculated the torque on each of the resulting elements independently of the other elements, under the quasi-steady assumption. We assumed that the aerodynamic force acts at the centroid of each wing element. The flow experienced at that point is due to the translation of the body in the direction of the jump (U) and the rotation of the wing about the pitch axis (⁠⁠), where r is the vector connecting the centroid of the wing element with the center of mass.

where A_n is the planform area of the nth wing element, is the angular velocity of the wing about the pitch axis and C_Fn is the aerodynamic force coefficient.

where K is interpolated from fig. S26, page 3-15, in Hoerner (1965, see also page 2-4). The forces calculated using the low estimate are expected to designate the minimal force production by the closed wings.

The total instantaneous pitching torque from the wings is obtained by summing the torques from the four wing elements in each movie frame. Substituting the result (τ_w) in Eqn 5 with τ_b (Eqn 8) and the rotational damping coefficient (Eqn 7) gave the total pitching torque in air (τ_pitch). With the mass moment of inertia, Eqn 5 can be used to predict the instantaneous angular acceleration (⁠⁠) of the body. To test the sensitivity of our model, we used the calculated estimates of torque to calculate the predicted angular acceleration of the body about the pitch axis. We used different values of I_yy for males and females based on intersexual differences in body mass and body length. We then averaged the predicted acceleration over time during the ballistic and wing supination phase in each take-off and compared it with the average angular accelerations measured from the movies. The latter was found by smoothing the instantaneous pitch angle data using a low-pass (300 Hz) Butterworth filter and finding the second derivative of the pitch angle with time (Rayner and Aldridge, 1985).

Throughout the paper, means are reported ±1 s.d.

In their resting position, the wings are pointing backwards along the sides of the insect, covering the dorsal and lateral sides of the abdomen like a gable roof (Figs 1, 2). The distal ends of the wings protrude a further 38±9.8% (n=10) of the body length beyond the posterior tip of the abdomen (Fig. 1). The wing center of area is located at 0.56±0.011 the distance from the wing base to the wing tip (wing length=0.91±0.125 mm), and coincides with the posterior tip of the abdomen (Fig. 1, n=16). Consequently, approximately half of the wing area and the maximal wing chord (located at 0.74±0.02 of the distance from the wing base to the tip, n=16) are located behind the posterior tip of the abdomen. The mean wing area was 0.58±0.072 and 0.39±0.079 mm² in females (n=20) and males (n=16), respectively. The supination of the wings (ε in Fig. 2A) in the resting position, as measured in the movies, was 31±7.2 deg (n=13).

The mean body length without the wings (B_L in Fig. 1B) was 0.81±0.106 mm (n=28), with a maximum width (lateral axis) and thickness (dorso-ventral axis) of 30±4.0% and 30±3.9% (n=10) of the body length, respectively. The body can be roughly divided along the longitudinal axis into two subunits: the head+thorax and the abdomen. The ellipsoids encompassing the two subunits are of similar volumes (ratio of 1.1:1; s.d.=0.32, n=10), suggesting that the center of mass is located at the junction between the thorax and the abdomen, which is located at 46±3.0% of B_L from the head (n=18). Females are slightly larger than males (m=41±11.9 and 25±5.7 µg, respectively, n=10 per sex; B_L=0.89±0.06 and 0.73±0.06 mm, n=15 and 13, respectively). As a result, the mean mass moment of inertia of females and males was 2.57×10⁻¹⁵ and 1.05×10⁻¹⁵ kg m², respectively. The average damping coefficient was calculated to be C_s=2.01×10⁻¹⁴ kg m² s⁻¹.

Out of 100 filmed take-offs, only nine showed wing flapping concurrent with the whitefly leaving the ground. In the remaining 91 (91%) take-offs, the whiteflies jumped into the air with their wings in the resting position. The hind legs were always the last to leave the ground. Fig. 2C shows the mean time allocation between the different phases in the take-off sequence. Wing flapping started in mid-air, 10.9±3.6 ms (n=74) after leaving the ground. The average ‘ballistic’ phase lasted 7.1±3.42 ms and wing deployment lasted 3.8±0.59 ms (n=73 and 81, respectively). By the time wing flapping started, the insects had reached a height above ground of 7.0±1.71 mm (i.e. 8.6 body lengths, n=34; Fig. 3). The mean speed at which the insects left the ground (take-off speed) was 0.61±0.097 m s⁻¹, and the average take-off angle was 64±10.7 deg (n=34). Table 1 summarizes the various kinematic parameters of the acceleration phase. The abrupt push against the ground resulted in a mean linear acceleration of 332.2±113.7 m s⁻² (i.e. 34 times gravity) requiring a (body) mass-specific power output of 105.5±45.7 W kg⁻¹ (n=27). When the whiteflies paused in preparation for jumping, their longitudinal axis was pitched upwards at an angle of 33±9.6 deg (n=32) relative to the horizontal plane (t₀ in Fig. 2B). During the acceleration phase, this angle decreased by 14±7.9 deg (n=32), suggesting a mean angular velocity of the body about the pitch axis of 22.9 Hz (i.e. revolutions s⁻¹, maximum observed=58.3 Hz). If the body had continued to rotate in the air at the same average speed, it should have rotated another 90 deg in the 10.9 ms between leaving the ground and the initiation of flapping (t₄−t₁), resulting in the insect moving through air upside-down. The whiteflies did indeed pitch forward after leaving the ground (Fig. 4A), but only by 34±24.9 deg on average (maximum=100 deg, minimum=2 deg, n=32). The pitch angle of the body (θ) at the start of the acceleration phase (at t₀) and at leaving the ground (at t₁) were positively correlated (linear regression, r=0.77, P<0.001, n=32), and the difference between the two angles (θ_t0–θ_t1) was negatively correlated with the minimum (negative) pitch angle reached in the air (r=−0.9, P<0.001, n=32). The minimum pitch angle reached in the air did not show a significant correlation with the pitch angle at t₀ (P=0.7), but did show a weak positive correlation with the pitch angle at t₁ (r=0.5, P=0.004, n=32).

Body rotation in the air abruptly slowed down and reversed direction during wing deployment (Fig. 4). However, a close examination of the curves in Fig. 4 revealed that in more than half of the trials (66%), body rotation slowed down to a halt even before wing deployment initiated. Within these trials, in 10 cases (31% of the total take-offs) body rotation even reversed direction prior to wing deployment. The mean angular accelerations differed significantly between the ‘acceleration’, ‘ballistic’, ‘supination’ and ‘elevation’ phases (one-way ANOVA for repeated measurements, P<0.001). The forward pitch rotation speed of the body increased during the ‘acceleration’ phase (mean acceleration −2.6×10⁶±2.16×10⁶ deg s⁻², n=32; see Fig. 4A for definition of rotation directions). The rotation speed slowed down and then accelerated in the opposite direction during the ‘ballistic’, ‘supination’ and ‘elevation’ phases (mean angular accelerations: 1.6×10⁶±0.83×10⁶, 3.6×10⁶±2.42×10⁶ and 0.5×10⁶±1.72×10⁶ deg s⁻², respectively). The mean angular acceleration during the ‘supination’ phase was significantly higher than during the ‘ballistic’ and ‘elevation’ phases (Tukey’s test, P<0.001).

Whiteflies with removed wings continued to rotate forward about the pitch axis after 15 ms from leaving the ground (Fig. 5A). Whiteflies with the distal half of their wings removed attempted to deploy their wings, but wing deployment did not result in the rapid backward pitch as in whiteflies with wings intact (Fig. 5B).

In insects with wings intact, supinating the wings in the resting position while the body moved through air (linear translation and rotation about the pitch axis) increased the projection of the wing area onto the flow (A′), as shown in Fig. 6A. Note that by the end of the supination phase (but prior to wing elevation) most of the wing area is perpendicular to the direction of motion in air. The calculated estimates from the biomechanical model (Fig. 6B,C) suggested that, once airborne, the aerodynamic torque acting on the closed wings was an order of magnitude higher than the torque acting on the body without the wings (Fig. 6B). Fig. 6B also illustrates the instantaneous aerodynamic pitching moments contributed by each of the four sections of the wings (BE01–04, see Fig. 1C). The two distal (posterior) sections of the closed wings together provided 89±2.3% and 87±6.9% out of the total torque from the wings for the high and low estimate, respectively. The total pitching torque (τ_pitch) became positive immediately at take-off, well in advance of the wing supination and elevation that occurred during wing deployment (Fig. 6C). However, during wing deployment, the magnitude of aerodynamic torque increased sharply (Fig. 6B,C), resulting in the sharp change in pitch angle of the body (Fig. 4B).

During the ballistic phase, the observed angular accelerations were significantly higher (paired t-test, P<0.001) and not different (paired t-test, P=0.46) than the predicted values derived from our model using the lower and higher estimates, respectively (Fig. 7A). The correlation between the predicted and observed accelerations was weak but significant (lower estimate: r=0.71, P=0.007; higher estimate: r=0.59, P=0.034). During the wing supination phase, the observed angular accelerations were significantly higher than both estimates (paired t-test, lower estimate: P<0.001; higher estimate: P=0.002; Fig. 7B) and there was no correlation between the observed angular accelerations and the angular accelerations predicted by the model (r<0.27, P>0.37 for both the lower and higher estimates).

Our whiteflies used only leg thrust to power take-offs in 91% of the trials, and the wings were deployed only after the insect moved several body lengths in the air. Weber (1931), who worked on greenhouse whiteflies (Trialeurodes vaporariorum), noted that the hind legs were capable of synchronized action to enable jumping. He further noted that the muscles powering the jumps were located inside the metathorax, and that whiteflies taking off from the underside of horizontal surfaces (e.g. leaves) fall 2–3 cm prior to opening their wings and flying. Our study, which focused on the tobacco whitefly (B. tabaci), corroborates Weber's observations and expands them to take-offs from the upper side of horizontal surfaces.

We hypothesized that in such take-off jumps, the rapid transition from ground to flapping flight would require one of three alternative take-off stabilization strategies: (1) the insect pushes against the ground with minimal body rotation, (2) rotations resulting from the push against the ground would be countered in the air by the flapping wings, or (3) the rotations would be countered in the air prior to wing flapping. Our measurements of timing of body rotation in the air (Fig. 4) support the third scenario, because the insects were capable of stopping body rotation prior to deploying and flapping their wings in 66% of the take-off jumps. In 31% of the jumps the insect even reversed the direction of body rotation prior to deploying their wings. However, it is also evident that wing deployment (supination and elevation) in itself provides a significant increase in torque, which reversed the direction of body rotation in the remaining trials, pitching the insect backwards into a posture more suitable for flight.

The acceleration phase was extremely short (1.7 ms on average) yet delivered high acceleration of the body (on average, 34 times gravity, power output: 105 W kg⁻¹ body mass; see Table 1). This supports the notion that B. tabaci utilize an elastic mechanism when jumping. Furthermore, the insects were immobile immediately prior to the acceleration phase, which may indicate that slow muscle work was taking place to store elastic energy in the body, locking some of the joints of the jumping mechanism and rendering the hind legs motionless.

One of the costs of a jumping mechanism based on abrupt release of stored energy is that it can send the body into the air rotating in an uncontrolled manner. Indeed, our whiteflies launched into the air rotating forward about the pitch axis. This occurred even when the analyzed take-offs were voluntary, executed from a flat and even surface, and the insects were uninterrupted as they prepared for take-off. Thus, the observed forward rotation of the body in the sagittal plane seems to be inherent to the take-off mechanism. The rotation likely resulted from the point of contact between the hind legs and ground being located posterior to the center of mass. As the hind legs pushed down against the ground, the resulting torque rotated the body forward in the pitch plane (Fig. 8). Once the body left the ground, the leg torque ceased to exist, but the body continued to rotate because of its angular momentum (Fig. 8B). Indeed, we found that the ballistic phase started with a high angular velocity (up to 58 revolutions s⁻¹, or 58 Hz) and that the minimum pitch angle achieved in the air was correlated with the rotation of the body during the acceleration phase.

Once in the air, during the ∼11 ms of air travel prior to the first downstroke, the body rotation decelerated and even reversed direction (Fig. 4). Because the insects were no longer in contact with the ground, any external stabilizing torque to resist the rotation of the body must have been due to aerodynamic force. During the ballistic phase, this aerodynamic force acts on the closed wings and on the rest of the body, but the quasi-steady blade-element model suggests that, by far, the closed wings provide most of the stabilizing torque (Fig. 6B). The fact that insects with removed wings continued to rotate in air (Fig. 5) supports this suggestion. Moreover, the model predicts that more than 87% of the stabilizing torque generated by the wings is generated by the distal half of the closed wings (Fig. 6B), which extends behind the posterior tip of the abdomen. Indeed, when only the distal section of the wings was trimmed, the insects did not stabilize body pitch during the jump. Hence, experiments as well as biomechanical considerations suggest that the wings play a substantial role in stabilizing body pitch before and during wing deployment. During the ballistic phase, the distal sections of the wings essentially operate as a posterior stabilizing surface resisting the rotation of the body. During wing deployment, the wings are rotated (supinated and elevated) to increase air resistance, further contributing to pitching the body backwards.

Such a function is performed by birds and man-made flyers using horizontal tails, located posterior to the center of mass, allowing some control over the pitch of the flyer. Insects do not have tails, and instead they use the forces generated by their flapping wings to dynamically stabilize their body orientation (Ristroph et al., 2013; Truong et al., 2014). Yet, when the wings are closed, dynamic stabilization is impaired. Ristroph et al. (2013) showed that the flight of fruit flies deprived of sensory data became unstable. The instability could be rectified by adding fibers to increase the drag on the posterior tip of the body, thus providing passive stability for pitch. Brackenbury (1996) noted that during jumping with the wings closed, the rotation rate of three homopteran species was low (5–9 Hz) compared with that of coleopteran flea beetles (35–187 Hz). He suggested that the relatively long wings of leafhoppers exert a stabilizing effect on body pitch when closed, by acting like ‘the tail-fin of an aircraft’. Our study of B. tabaci provides evidence for passive stability by the close wings during the ballistic phase. In more than half of the cases, the insects were able to stop the high angular velocity of the body even while the wings were in the resting position. The area of a wing pair is 3.2-fold the planform (in the frontal plane) area of the body. Therefore, aerodynamic forces on the large flat wings can greatly exceed air resistance on the elliptically shaped body. Furthermore, because of the shape and size of the wings, the maximal wing chord and half of the wing area are posterior to the tip of the abdomen (Fig. 1), so that the aerodynamic force on the distal ends of the wings can result in substantial pitching torque at the center of mass. As the wings supinated, the insects rotated and the wings elevated, the area of the wings projected onto the flow increased, and so did the aerodynamic torque that countered the forward rotation of the body (Fig. 8).

The simplistic biomechanical model underestimated the angular accelerations observed during the supination phase (Fig. 7B). On average, the angular acceleration predicted by the model using the higher estimate was only 63±34.8% of the observed value (n=13) and there was no correlation between the predicted and observed values. This may be due to the fact that wing supination was much shorter (1.6 ms; Fig. 2) and immediately followed by wing elevation. Thus, internal torques, resulting from small movements of the wings and abdomen relative to the thorax, may be at play during the end of the supination phase, and these were unaccounted for in our model. Indeed, the wing elevation phase was associated with a rapid pitch of the body backwards (Fig. 4). Wing elevation may contribute to rotating the body in a second mechanism that is not accounted for in our aerodynamic model. Namely, the inertia from rotation of the wings (and added mass of air) about the hinges in the pitch plane can result in counter-pitch rotation of the body backwards. The delicate, small wings of whiteflies precluded the measurement of mass distribution along the wing and therefore wing inertia is not accounted for in our model. However, the effect of wing inertia and internal torques, if substantial, would only apply at wing deployment and will only add to the aerodynamic effect already described by our model.

The wing tips of many homopteran insects substantially extend behind the posterior tip of the abdomen, and some of these insects are also known to be prominent jumpers. It would therefore be interesting to examine how common this longitudinal stability mechanism is in other insects that power take-off with their legs while the wings are in the resting position.

We thank J. Byres and S. Naranjo from the USDA, Western Region Biomass Research Center, for providing body mass measurements of male and female whiteflies. We thank N. Paz for English editing and A. Ayali for suggesting improvements to the manuscript. We also thank the staff of the I. Meir Segals Garden for Zoological Research for logistical assistance.