Air Turbulence and Tube Currents

Some sources of aberration have nothing to do with the telescope itself. They come from the necessary immersion of the instrument in a changing optical medium. The light of astronomical objects must traverse a turbulent column of air that extends many kilometers from the top of the atmosphere to the focal plane of the instrument.

Little can be done by the amateur observer about turbulence high in the atmosphere, but it is easy to recognize in the star test. Many of the problems described in this chapter cure themselves after a time, particularly those having to do with the cooling of the telescope or of its immediate environment. The aim here is to teach you how to detect aberrations that originate in the motion of air, prevent them as much as possible, and recognize when they subside. Learning the star-test behavior of the atmosphere at your location will also allow you to identify those rare bouts of unusually steady seeing favorable to high magnifications.

7.1 Air As a Refractive Medium

The index of refraction of air is very near that of a vacuum, but it is noticeably different. At 0° Celsius, dry air at sea-level pressure has an index of refraction of 1.00029, while vacuum has an index defined exactly as unity (CRC 1973). Such a small difference doesn't seem worth worrying about, but the wave going through the air inside a 1.5-meter focal length telescope is slowed, relative to passage through vacuum, by about 791 wavelengths.

Assuming that air is approximately an ideal gas and that the fractional part of the index of refraction changes linearly with temperature, one can easily figure out the lag for a small temperature difference. Recall that 0° Celsius is 273° Kelvin in units of absolute temperature. Therefore, a 1° K difference in temperature over a distance of 1.5 meters results in a delay of 791/273 wavelengths per degree Kelvin, or about 2.9 wavelengths/0K (1.6 wavelengths/0 F). Now stretch the tube of air up through the atmosphere, many kilometers overhead, with each layer at a different temperature. Propagation delays can be profound.

However, it doesn't matter if the pencil of light is delayed uniformly. After all, the light has been on a long journey. Who cares if it arrives a little late? We are only interested in the variations in direction or time of arrival as the light enters different portions of the long, skinny cylinder of air in front of our instruments. We detect those differences when we see the image degraded by bad seeing. When we look through Earth's atmosphere, we hope for a pressure and temperature uniformity that does not often exist. Some parts of the wavefront are a small distance behind other portions of the wavefront. Such aberrations can form differences in intensity and apparent location (i.e., "twinkling"), but their most common effect on large apertures is to blur the image.

In defense of the sky, perhaps we are expecting too much when we peer upward through all of that material and demand perfect images. After all, the total pressure of the atmosphere is the same as the pressure of over 30 feet of water. Sub-arcsecond resolution would not be expected from the bottom of a diving pool. Yet such resolutions actually are witnessed for images seen through the atmosphere. On exceptional nights, the atmosphere surprises us by becoming beautifully tranquil.

Still, a mechanism producing small-scale differences in refractive index must exist before atmospheric effects become troublesome. Air, by its gaseous nature, doesn't tend to maintain differences in pressure or temperature—except for wide layering caused by the force of gravity. Air mixes together, averaging differences until layered uniformity prevails. Statistical inhomogeneity won't persist without mixing. The two mechanisms of most interest to telescope testers are atmospheric turbulence and tube currents.

7.2 Turbulence

If the atmosphere changed slowly, turbulence would never start. However, the atmosphere often is forced to move quickly. As sunlight deposits energy on the ground, it heats the air immediately above it. That air expands, becoming lighter than the mass of air immediately above it. The situation becomes unstable or "active," and the denser air falls to replace the warmer air below it. It moves quickly enough to generate turbulence.

This type of fluid motion is called a "Rayleigh-Taylor" instability. Completely fill an empty soda bottle with water and invert it with a playing card over the opening. Carefully remove your hand. The water does not fall out of the container (Walker 1977). If you were to suddenly remove the card, you would have a bottle full of water poised over a space of air.

All the water can't fall immediately. Air pressure holds the water up in the same way that a column of mercury is held up in a barometer. All of the water molecules are holding on to each other so they won't fall away individually.

Faster than humans can perceive, the random jiggles of the surface will cause one portion to deform slightly upward and another part slightly downward. That's all it takes. Once this process starts, it drives itself. A bubble rises and finally breaks off. The bottle empties, but it doesn't do so uniformly. The instability must form again and again. The bottle drains noisily, gently kicking in your hand.

Incidentally, the instability doesn't take place if a sufficient external force holds the surface level. With the card, the surface was constrained by the structural strength of the paper. By the time the opening is reduced to the size of a soda straw, surface tension alone provides enough force to overcome the instability. You can lift narrow columns of water by merely plugging the top of a straw.

For the atmosphere, no bottle circumscribes the unstable region, but essentially the same process occurs. As the cold air falls, inefficiencies ensure that the edges of the falling region aren't smooth. The fall is quick enough that tiny vortices form, but smaller swirls don't move with the speed of the main convective cell. Even tinier vortices form at the edges of these eddies. Finally the swirls become lost in complexity. At some small scale, the model of collective motion breaks down and the energy is expressed in heat. All realistic cases of fluid flow, such as streams and large convective cells, move macroscopically only on the average. When considered microscopically, they are turbulent.

7.2.1 The Aberration Function

Statistical variations of the wavefront are usually handled with semi-analytic procedures. These procedures, which commonly assume a Gaussian form for the random variations, are quite useful for calculating the longexposure MTFs and other features associated with rough surfaces (Schroeder 1987, p. 315). However, they do not allow us to calculate the appearance of a single example image. They are the behavior averaged over many such roughened apertures.

The method used here to simulate the wavefront is called the midpoint displacement fractal algorithm. It is used to generate wonderfully realistic fractal landscapes (Peitgen and Saupe 1988, p. 96; Mandelbrot 1983; Harrington 1987). It can also render an artificial, pseudo-random wavefront just after it has passed through the aperture. Fractals were first applied to diffraction problems by M.V. Berry in the seminal article "Diffractals," in which a fractally-derived phase screen first appeared (Berry 1979, 1981). The method used here is an adaptation of an earlier article, where a one-dimensional variation of the midpoint deviation algorithm was used to calculate slit-type diffraction patterns (Suiter 1986a).

Because of the way the fractal must fill in the area of a two-dimensional grid, we may conveniently proceed by a two-step iterative procedure. The algorithm starts with the four corners of 129 x 129 point grid assigned arbitrarily to be of height zero (Fig. 7-la). Only the location of the points are shown in Fig. 7-1; the deviation is perpendicular to the paper.

For the first half of the first iteration (Fig. 7-1b), the offset of the next finer division is calculated as the average of the previous four set points at the ends of the dotted curves (for the first iteration, this average is zero) plus or minus some random deviation (±Az). The center point is assigned this value. The points set only during each half-iteration are black; all previously assigned points are white. Points of the 129 x 129 grid that have yet to be assigned values aren't shown.

During the second half of the first iteration, the edges are filled in. The four nearest points are averaged again, assuming that the points off the edge of the area are zero, and to this number is added another plus or minus deviation. This time the maximum allowable deviation is divided by V2 before it is assigned. Notice that each complete iteration cycle averages the positions of the points first oriented at the diagonals to the point to be set, and then in the second half, averages at rectilinear angles. For obvious reasons, this algorithm can also be called the "x+" method.

At the start of the second iteration, the maximum allowable offset is divided once again by V2 to make it ±Az/2. At the beginning of the third iteration, it is ±Az/4, and so forth. Thus, the cell sizes are decreasing at precisely the same rate as the maximum deviation, setting up conditions so that the apparent random slopes average out to be the same at all scales. This is called statistical self-similarity, or an independence of scale for averaged behavior.

Finally, when the seventh iteration is completed (it would be the 15th frame of Fig. 7-1, if the figure were allowed to go that far), the entire 129 x 129 grid is assigned. At this point, with the fractal algorithm finished, the area is conditioned to resemble a circular aperture. The aberrations and transmissions of points farther than 64 locations away from the center point are set to zero, and if a secondary is simulated, the same is done for all points within a certain radius. The outer parts of the 129 x 129 grid is cut off like the extra dough of a pie crust. The statistics of the clear aperture are then calculated, and the RMS deviation is scaled to the value

Fig. 7-1. The ordering of point assignments of the midpoint deviation algorithm. Here the roughness is out of the paper.

demanded by the individual computation.

An easy misunderstanding of Fig. 7-1 could occur here. Because of its superficial resemblance to the later image frames, readers could incorrectly assume that this non-physical mathematical procedure is used to calculate an image. This method merely simulates the random aberration function on the pupil, not an image. The image is calculated with the same Huygens-Fresnel theory used in other chapters (see Appendix B).

Physical reasons may determine that random variations do not persist equally over all scales. In fact, such is the case for turbulence. The characteristic width scale of turbulence, or about the distance between the "bumps," is on the order of 2-20 cm, with a good one-number estimate of 10 cm (Roddier 1981, p. 302; Schroeder 1987, p. 314). Abrupt deformations of the wavefront for slight lateral motions are not anticipated. The roughening of the wavefront is slightly rounded. This feature results in the well-known "small telescope" effect, where tiny instruments seem to give better images than large ones. Apertures smaller that 100 mm are looking through portions of the wavefront that are closer to a plane. Turbulence causes the image to jump around, but it appears to be well delineated from moment to moment.

For this reason, the algorithm is slightly modified. A quenching factor is applied to the V2 diminishing of each half iteration. The divisor in this case becomes V2 q, where q is the quenching factor. For q greater than 1, the surface softens to make the gross variations more noticeable, with fine-scale roughness relatively suppressed. The surface resembles crumpled smooth paper instead of sandpaper. The quenching factor used here was between 1.1 and 1.6. No attempt was made to physically justify this model, but the lessening of finer scale eddies demands some sort of smoothing. The images generated with such quenching factors look the most authentic, and that realism justified their use in the images that appear here.1 For quenching factors different from 1, the surface is no longer self-similar.

The other modification is a way of controlling the largest scale of the roughness. We would expect turbulence to show itself differently in a 24-inch behemoth than in a tiny refractor. In large telescopes, the image does less jumping around and momentary clearing of the blur happens less often. For the purposes of this chapter, the largest scale of the roughness was set at a significant fraction of the aperture. Thus, the calculations below are good for small telescopes with aperture of about 200 mm for most cases of turbulent air.

Some important characteristics of this fractal aberration will affect the quality of our simulation. First, the surface distribution is not Gaussian (bell-curve shaped). The surface distribution (shown in Fig. 7-2) is roughly Gaussian in that it is more-or-less peaked, but under no circumstance is it Gaussian nor would it ever be Gaussian, even if the iteration were allowed to continue forever. More commonly, multiple peaks are found in the surface distribution. Such unusual action allows us to simulate detail that would be washed out of Gaussian models.

Second, the surface is locally correlated. A surface with memory will allow our images to contain realistic streaks and bumps. The MTFs appearing below are calculated with the modeled "snapshot" surfaces and will show statistical variations. This procedure gives us an important view that could not be attained if we figured the MTFs from long-exposure averages.

One feature compromises the algorithm described above. The fictional exterior points are assumed to be zero, so one might expect some unusual distortions to occur near the edges. Because the surface was pie-trimmed, the worst effects are toward the four compass directions of the pattern nearest those edges. Distortions were indeed seen, but the problem areas were small and the effects did not readily appear in diffraction patterns.

1 Readers interested in the more physical time-averaged models can find a survey and a good bibliography in the review article by Hufnagel (1993).

Surface height distributions used in turbulence calculations

1 RMS height

Fig. 7-2. Example wavefront-height distributions of the turbulent wavefronts, with the x-axis measured in units of the roughness RMS value.

Fig. 7-2. Example wavefront-height distributions of the turbulent wavefronts, with the x-axis measured in units of the roughness RMS value.

Fig. 7-3. An example modeled aberration function of air turbulence.

An example turbulent wavefront is shown in Fig. 7-3, with the wave-front conveniently elevated a small distance through the aperture. The most pleasing aspects of this pattern are the pseudo-random creases running through it. Creases should model the shadow-band behavior of real turbulent air currents that cause speckles and temporary spikes in the image. The quenching factor has acted to smooth the fine-scale variation, which will become more apparent when compared to the aberration function for primary ripple in the roughness chapter. Because earlier iterations were capable of much more movement, occasional dimples appear in the aberration function.

Filtration caused by turbulence

Filtration caused by turbulence

Fraction of maximum spatial frequency

Fig. 7-4. Twelve MTF curves associated with 0.15 wavelength RMS air turbulence. Aberration functions are calculated with the fractal model described in the text.

Fraction of maximum spatial frequency

Fig. 7-4. Twelve MTF curves associated with 0.15 wavelength RMS air turbulence. Aberration functions are calculated with the fractal model described in the text.

7.2.2 Filtering Caused by Turbulence

These apertures are not circularly symmetric. Their ability to preserve contrast depends on the orientation of the bar pattern of an MTF target. Therefore, the MTF for each of the four generated surfaces was calculated along 3 axes. All 12 such MTF curves are shown in Fig. 7-4. The amount of RMS aberration used was about twice the 1/14-wavelength Maréchal tolerance. If one looks at Fig. 7-2 and replaces the RMS height with 0.15 wavelength, it is apparent that the total wavefront aberration is about 4 times that value, or 0.6 wavelength. This aberration is about twice as bad as can be tolerated for highresolution observing, but such aberration is by no means uncommon for air turbulence. Often, seeing is much worse.

Also notice the extreme fluctuation at the high-frequency end of the chart. Because the curve flutters around rapidly there, resolution is limited to about 1/2 to 2/3 of the theoretical maximum for the aperture. With a 200-mm aperture, the central blur circle has a radius of about 1 to 1.5 arcseconds.

7.2.3 Observing Turbulence

In the focused image plots of Fig. 7-5, the modeled turbulence corresponds to a 5 on Pickering's 1-10 seeing scale, since the focused disk is always visible but arcs aren't often seen. This number corresponds to a

TURB = 0.15 waves RMS LATER normal OB=20% 10

TURB = 0.15 waves RMS LATER normal OB=20% 10

Fig. 7-5. Image patterns calculated for 0.15 wavelength turbulence. Perfect patterns are to the right. Central obstruction is arbitrarily set at 20% of the aperture. (For a description of the labeling of image figures, see App. D.)

"poor" seeing rating (Muirden 1974). Sometimes turbulence is much more severe. Good lunar-planetary viewing requires better.

Fig. 7-6 tracks the focused image as turbulence aberration becomes less objectionable. Fig. 7-6a shows long arcs and probably fluctuates between Pickering ratings of 6 and low 7. Fig. 7-6b has a rating of high 8, since the rings are complete but are always moving. Fig. 7-6c is about a high 9 or low 10, since the rings are stationary and the disk is crisply defined, but the weak ring still breaks up. With as much as '/20 wavelength of turbulence wavefront deformation, seeing still has a 10 out of 10 rating.

a) 0.10 waves RMS b) 0.075 waves RMS c) 0.05 waves RMS 10

a) 0.10 waves RMS b) 0.075 waves RMS c) 0.05 waves RMS 10

Fig. 7-6. Focused images as turbulence is lessened. The ring .structure reappears.

Fig. 7-6. Focused images as turbulence is lessened. The ring .structure reappears.

The turbulence aberration is easily distinguishable from other aberrations:

1. It moves quickly. In less than a second you see an entirely different pattern.

2. It is balanced on either side of focus (unbalanced aberrations can modify the patterns, but the behavior is not caused by air turbulence).

3. If you pull the eyepiece outside of infinity focus, you can often focus on the disturbances themselves high in the atmosphere. They appear as bands or cells moving across the out-of-focus image.

7.2.4 Corrective Action

You can do little about high altitude turbulence, since it is beyond reach. High turbulence is more a function of the climate rather than a local phenomenon. However, you can start a log of weather and seeing conditions and see if you can come up with correlations. In general, the presence of clouds and high wind indicates that surplus energy is being transported around in the atmosphere and that seeing is bad. Good seeing is not always associated with transparent nights and may in fact be negatively correlated. Tranquil nights tend to be a little hazy.

Local seeing, or turbulence that occurs within a few hundred feet of the ground, is another matter. Local air turbulence may be caused by thermal currents from buildings or structures that have yet to cool from daytime heating. House shingles are notorious for their long cool-down times. Paving asphalt also retains heat and gives it up slowly. For this reason, observing over grass or trees is much preferable to observing over houses or roads.

Some authors have remarked on the interference caused by ground seeing (Muirden 1974), which is an effect located very near the telescope.

Personally, I have never had any trouble with turbulence very near the telescope that was not caused by setting the telescope directly on asphalt or concrete. One exception to this general situation, however, is that the nearby observer is a very good furnace. Body heat can waft across an open tube quite easily. This problem is not too great in the summer, when temperature differences are lower, but in the winter it can do serious harm to an image. A cloth drapery for an open tube framework often helps here.

One thing to watch when several people are observing together is that those who are waiting to look don't congregate near the optical path of a telescope, or upwind of it. Heated air from their bodies or breath can intercept the incoming light beam. If possible, when hosting a public observing session, arrange the line on the down-breeze side of the optical path. Finally, if you have to transport the telescope to the site, be sure to park the automobile or truck so that air rising from the hot engine cannot interfere with anticipated highresolution observation.

7.3 Tube Currents

Air at different temperatures is affected by gravity because cooler air weighs more. When unconstrained by exterior structures, it forms the convection cells discussed above. Air inside a tilted tube tends to follow the wall—hot air on the high side, cool air on the low side. The tube resembles a tilted stovepipe. As air is heated at the bottom and becomes less dense, cool air falls down the tube and forces the hot air upward. It rises to hug the tube on its upper side and eventually exhausts to the outside.

With a telescope at or near ambient temperature, temperature differentials aren't bad enough to cause tube currents. However, when a telescope is first taken outside, thermal inertia causes problems until the instrument reaches the environment's temperature. The thick glass of the objective is particularly prone to slow cooling.

7.3.1 The Aberration Function

Clearly, every telescope cools off differently. Some have other problems that may obscure or modify the tube currents, like a hot mounting or an observing pad that retains heat from the day. Some have peculiarly shaped tubes or partial tubes that would change the patterns modeled here. All cooling telescopes have an unavoidable amount of locally induced turbulence. Some telescopes have only stubby mirror boxes and no real tubes, and these generate different patterns than shown below.

Schmidt-Cassegrains and refractors must cool off only through their rear exit portals or directly through the tube by radiative and conductive cooling. One Schmidt-Cassegrain I examined displayed an extension of the secondary shadow on one side of focus and streaks parallel to the edge of the secondary on the other. At first, I thought it had a cracked corrector plate near the secondary mounting hole, but when the tube orientation changed, the pattern always followed an up-down direction. Presumably, a great deal of the cooling was taking place in or around the mirror or corrector plate perforations. An uneven thermal effect of the Cassegrain baffle tube may also have caused the problem.

Even though it is not matched by every telescope, the behavior modeled below is common enough in small to medium-sized Newtonians with round tubes. The model assumes that the warm air is confined to the upper side of the tube and that its effect is to advance the wavefront only along that upper side, leaving the rest of the optical path comparatively untouched. This behavior was described in Chapter 2 as resembling the turning of a page.

The model used here is

where x is the linear coordinate in one direction across the surface with the origin at the center of the aperture. The value of x reaches 1 at the edge of the aperture.

The model does not allow variation in any other direction than the up-down coordinate. No roughness is superimposed on the effect of the tube current, even though it would surely be present.

The aberration plotted over the pupil is shown in Fig. 7-7, with the up direction toward the right.

7.3.2 Filtering of Tube Currents

Again, the value of the modulation transfer function depends on the orientation of the bar pattern. The function was calculated for three angles: up-down, left-right, and a 45° tilt. Fig. 7-8 shows two cases. The 1/2 wavelength example is as bad as an observer should tolerate. The 1 wavelength case is severe, but not unusual for telescopes that have just been moved from warm surroundings.

This transfer function plot possesses a number of interesting features. The first is the sudden drop of both the 45° and the horizontal MTFs. The sharp fall is caused by the localized nature of the aberration. The steep slope of the aberration kicks a lot of light out of the diffraction spot, affecting spatial frequencies even 1/10 of the maximum. For example,

Fig. 7-7. The tube-current aberration function modeled over the aperture pupil.

Filtration caused by tube currents

Filtration caused by tube currents

Fig. 7-8. The MTF for the modeled tube current aberration. Two different aberrations are shown. Each aberration has three curves, but the up-down bar orientation results in no degradation. The up-down MTFs of both aberration values are plotted identically on top of the perfect pattern.

Fraction of maximum spatial frequency

Fig. 7-8. The MTF for the modeled tube current aberration. Two different aberrations are shown. Each aberration has three curves, but the up-down bar orientation results in no degradation. The up-down MTFs of both aberration values are plotted identically on top of the perfect pattern.

if you had a 200-mm aperture, resolution of 5-arcsecond details would be noticeably degraded.

Second, for one curve in each aberration amount, the contrast is unaffected. The smearing of the image in the vertical direction doesn't affect resolution of MTF targets with the bars oriented up-down. Of course, the unmodeled roughness would tend to break up this symmetry a little.

7.3.3 Observing Tube Currents

Tube currents are easy to see. The problem lies in determining whether or not the aberration is in fact caused by a tube current or is present in the glass. These currents can be remarkably stable. You would think that they would dance and sway like high-altitude turbulence. The patterns do change, but they do so slowly, like candle flames. Even though you may know intellectually that a candle is a dynamic process, when you look at the fire, you easily slide into the comfortable viewpoint that it is motionless. A candle flame seems perched atop the wick.

Truly hot telescopes boil with turbulence, but they don't show this effect very long. As the telescope cools, the tiny temperature differences do not support the formation of massively turbulent air masses. The air moves slowly and smoothly upward. It drifts side-to-side languidly, but at any given time it is relatively quiet. Tube currents are always influenced by gravity and hence are oriented in an up-down direction.

You may easily determine the orientation of a suspected tube current wavefront deformation for refractors and Cassegrain-style instruments, just don't use a right-angled bend in the optical path. For a Newtonian reflector, however, determining the angle of the image is not straightforward. The cause is, of course, the built-in diagonal reflection.

Two remedies are suggested. The first is to use a star on the north-south meridian. Direction can be determined from the western drift by turning off the clock drive (if one is used). This trick cannot be used for an artificial source test. For such a telescope, a simple expedient is to rack the eyepiece far out of focus and then insert a fist or other obstruction from a known angle in front of the aperture. Up or down can then be easily located.

The diffraction pattern calculated for a total aberration of 1 wavelength appears in Fig. 7-9. The pattern is squeezed-in on one side of focus and stretched-out on the other.

Once you see a tube current, make certain that some other difficulty isn't the problem. First, change the tube orientation. Locate a feature of the tube along which the stretching is pointed. If no tube landmark exists, make a slight mark on the tube or attach a little curl of tape. Then, rotate the tube by some reasonable angle. Unfortunately, fork-mounted Schmidt-Cassegrains are impossible to rotate. Use a test source at a different location.

Tube currents will still point up and down with the new orientation, but they stretch toward a different tube feature. Other difficulties, such as warped or damaged optics, now show a non-vertical tilt in the eyepiece.

tube current OB=30% 10 normal OB=30% 10

tube current OB=30% 10 normal OB=30% 10

Fig. 7-9. The star test patterns of 1 wavelength of tube current aberration. The perfect aperture is in the column on the right.)

Fig. 7-9. The star test patterns of 1 wavelength of tube current aberration. The perfect aperture is in the column on the right.)

7.3.4 Corrective Actions for Tube Currents

Tube currents are not really that serious for very small telescopes. Just wait until the telescope cools down to the ambient temperature. If environmental temperature varies so much that the telescope never really catches up to the local temperature, atmospheric conditions are so unstable that seeing will be poor anyway.

Sealed telescopes, particularly refractors, do not display the same tube current effects that are common in open-tube reflectors. Refractors are typically made with metal tubes that leak heat readily, so they quickly cool

Chapter 7. Air Turbulence and Tube Currents down. Setting up small telescopes in the late evening, with enough time to reach the ambient temperature, is often sufficient to eliminate this optical problem.

The only observers who may find tube currents to be both objectionable and persistent are those using large or especially thick telescope objectives. I once worked on a 16-inch Newtonian mirror of 3-inch thickness (1:5 ratio). This mirror required half the night to cool down even under relatively benign conditions. Many nights it never cooled, but when it did, the mirror performed magnificently.

Such mirrors really should be prepared before they are required for observing. Transport in a warm automobile is one of the worst things that can be done to a thick mirror. Almost equally bad is storage in a sunlit shed or observatory. The optimum procedure is to open up the telescope or observatory in the early evening, long before the instrument is required. If the telescope is transported, try not to carry it in the heated passenger area of an automobile. Instead, haul it in a trailer or in the back of a truck.

Sometimes, the telescope must be set up on concrete or asphalt. Even with a properly stabilized instrument, the tube can catch external currents and act as a duct for them. In such situations, try closing the bottom end of a Newtonian's tube with a plastic bag and see if the aberration decreases.

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