Correction of image bloom in trail widths using a stellar calibration curve

To correct image bloom from the system, a two-dimensional isotropic Gaussian was fit to stars that were stationary in the narrow-field system. Stars characterised the image bloom of the system as they are point sources broadened by the system PSF. These stellar Gaussian fits were then used to produce a calibration graph, like Figure 3.6, plotting star width versus peak brightness. Measured meteor trail widths for each slice were corrected by subtracting the width of an equivalent peak brightness star, determined from the calibration graph, from the raw meteor trail width.

The hyperfocal distance of the narrow-field cameras, or the distance past which all objects are in focus, provided the system is focused at infinity, was calculated to ensure that meteors were equivalently in focus as stars, and that bloom corrections derived from observations of stars could be applied to meteors. For the narrow-field camera, with aperture sizeD= 80 mm (effective 50 mm), f/6.8 (N = f/D= 6.8), and pixel size 7.8µm setting an upper bound to the

blur radiusc, the hyperfocal distance, H = ND _D c +1 , (3.2)

is 5.9 km. Thus, objects at a range beyond 5.9 km, including all meteors studied, where the average range was over 100 km, were equally in focus as stars observed with the narrow-field camera.

A Gaussian is expected to represent the PSF of the system provided that the most sig- nificant effects were contributed by the telescope (diffraction and chromatic aberration, as in Hawkins & Whipple (1958)), as well the CCD (charge leakage between neighbouring pixels, as in Groom et al. (1999)). The Gaussian PSF is isotropic as the pixels of the CCD are square, and contributions to the PSF from diffraction are expected to be a function of radial distance only. Later, in Section 3.4.1, it will be demonstrated that most of the observed Gaussian PSF appears to originate from the CCD or intensifier: a transient, shot noise artefact produced by the intensifier is observed to have a Gaussian brightness profile that is completely removed by bloom correction.

The form of the fit is given by

B(x,y)= a+(b−a) exp " −(x−cx) 2+_(y−c y)2 2d2 # , (3.3)

with B(x,y) representing the brightness of pixel (x,y), abeing the background mean level, b

being the peak brightness, (cx,cy) representing the centrex- andy-pixel of the star, anddbeing the spread or variance. The width of the Gaussian to threshold brightness Bt may be found by inverting Eq. (3.3). A quick way to get the star width from Eq. (3.3) is to solve for the two roots x₊and x− along the liney = 0 for the brightness equivalent to the threshold brightness,

Bt. As the profile is isotropic, the width of the Gaussian star profile is simply x+− x−, giving

Table 3.3: Parameters for stellar width calibration curves for each night of observation.mcalib,01 andbcalib,01 are the calibration slope and intercept for Tavistock, whilemcalib,01 andbcalib,01 are for Elginfield.

Date mcalib,01 mcalib,02 bcalib,01 bcalib,02 20101020 17.5±0.2 20.4±0.4 −52.0±0.8 −53±2 20101103 18.41±0.07 21.1±0.2 −54.8±0.3 −59±1 20101106 21.9±0.3 20.9±0.3 −71±1 −56±2

The calibration graph, Figure 3.6, plots the squared star width to the threshold brightness of the mean background plus two standard deviations, versus the natural logarithm of the star’s peak brightness minus the background brightness. The result is a linear fit,

w2_star =mcalibln(b−a)+bcalib, (3.5)

with a slope,mcalib, equivalent to 8d2, and an intercept,bcalib, of−8d2ln(Bt−a). In the special case of the star FWHM,Bt =(b+a)/2 and Eq. (3.5) becomes

w2_star,FWHM =mcalibln 2. (3.6)

Stellar calibration graphs were produced for each night of observation, for each station. This was necessary due to changes in the slope and intercept of the curves over each night, as shown in Table 3.3. This was likely due to changes in temperature, humidity, and other conditions in the system or outside.

Raw trail widths were corrected by substituting the peak brightness of the measured meteor profile, as well as the background mean brightness, to the stellar calibration curve to get the width of a star with an equivalent peak brightness, wstar. This star width was subtracted from the raw trail width to give a corrected trail width. A similar method was employed by previous optical studies (Hawkins & Whipple 1958, Cook et al. 1962, Kaiser et al. 2004) to correct for

image bloom. The corrected measurements are given by

wobs,corrected =wobs−wstar, (3.7)

wfit,corrected =wfit−wstar, (3.8)

wFWHM,corrected =wFWHM−wstar,FWHM, (3.9) wherewobsis the width of the observed brightness profile 2σabove the background brightness,

wfitis the equivalent width for the Gaussian fit to the observed brightness profile, andwFWHMis the FWHM of the Gaussian fit.

Uncertainties in the slope and intercept of the fit were found by using the bootstrap method described by Press et al. (2007). This method produces simulated data sets of equivalent size to a measured data set by randomly selecting points from the measured data set, allowing duplica- tion. By producing fits to the simulated data sets, a distribution of fit values emerge. This allows for determination of the mean fit values, as well as associated uncertainties derived from the standard deviation of the simulated fit values. This method was particularly appropriate here, as the distribution of star brightness (and width) from frame to frame was not explicitly known, but could be implicitly estimated with this method, and converted to a distribution for mcalib andbcalib.

The uncertainty in the fit slope and intercept were used to calculate the uncertainty in the star width, ∆wstar = ∆(w2 star) 2wstar , (3.10)

wherewstar is given by Eq. (3.5), and

∆(w2_star)= q

meteor velocity vector line of sight vector (from meteor to camera) perspective angle p narrow-field camera meteor

Figure 3.7: Illustration of the perspective angle.

3.3.4

Conversion of measurements from pixels to metres, effect of image

smear, and final trail width uncertainty

The corrected trail widths, wcorrected, measured in pixels, were converted to final trail widths,

wfinal, in metres, by making use of the range, R, from the narrow-field station to the meteor from the trajectory solution,

wfinal =wcorrected· pscale·R. (3.12) The angular scale per pixel for the narrow-field system, pscale, was also used. Similarly, the pixel distance along the meteor trail for each width measurement,d, was converted to metres,

dfinal =d· 1 sinθp

· pscale·R. (3.13)

An additional corrective factor, 1/sinθp, was employed to compensate for the perspective of the narrow-field station when viewing the meteor. Figure 3.7 illustrates the angle,θp, between the meteor velocity vector and the line of sight vector.

Trail smearing occurs if the meteor drifts, frame-to-frame, in a direction that is not the trail direction, due to imperfect tracking with the narrow-field system. In that case, θtrail ,

θtracking, where the tracking angle is a function of frame, since it varies as the meteor is tracked. Smearing is also dependent on the apparent distance that the meteor moves in each frame. The

frame-to-frame tracking length,ltracking, measures the difference in the meteor’s position in the frame from one frame to the next. ltrackingwas calculated using a second-order polynomial fit to the stored positionsx(f) andy(f) of the meteor, found using the pattern recognition algorithm described in Section 3.3.2,

ltracking(f)= p

[x(f +1)−x(f)]2+_[y(f +_1)−y(f)]2. _(3.14)

The derivatives of x(f) and y(f) were also calculated, analytically, with respect to frame and used to determineθtracking,

θtracking(f)= tan −1 dy dx _f = tan−1 dy/d f 0 dx/d f0 f0_=f . (3.15)

The resulting uncertainty due to trail smearing was

∆wsmear=ltrackingsin|θtrail−θtracking|. (3.16)

The two other sources of trail width uncertainty are variation in the raw width due to pixel interpolation in the original event video, and uncertainty in the width of stars used to correct the raw widths. The total uncertainty in the final, corrected width (in metres) was thus found to be

∆wfinal = pscale·R· q

∆w2

raw+ ∆w2star+ ∆w2smear. (3.17)