Half Flux Diameter

-Applicate to determination for faint star event-

Kazuhisa Miyashita

19 Aug 2007

Revived 1 Oct 2007

1. Introduction

It is difficult to determine the star appears or not in the faint star event, because, in the case of noisy video, the star often disappears into large noises. Half Flux Diameter function of Limovie is useful for analyzing in such cases. In this paper, I explain what means Half Flux Diameter is, and how to use it for timing determination of occultation event.

Figure 1　Half Flux Diameter

In figure 1, the circle which has diameter same as Half Flux Diameter is indecated red line to see easily. (Note:In Limovie's image area, the Aperture is indicate by red circle.)

2. Explanation of Half Flux Diameter(HFD)

Half Flux Diameter is defined :

The diameter of circle where total of brightness value of pixels inside it is equal outside pixel's. (It is in the Aperture region.)

The “brightness value of pixel” is the value substructed the mean of background area.

The parameters are ..

Vi: brightness value of each pixel

di: distance from gravity centroid to each pixel

H : Half Flux Diameter

The relation of these parameters is explaned by the formula.

・・・(1)　　　then ・・・ (2)

Hence, the HFD is ..

(3)

The denominator is same as “aperture value”, and the molecule can be calculated fastly.

A complex and time-consuming fitting, which is necessary to obtain FMWH, does not need doing. It means that HFD is very suitable for the software which contimuously processes a lot of images(frames) like Limovie.

3.The expected value of Half Flux Diameter

Figure 2 HFD in the sky background area and on the star.

A is the diameter of measurement aperture. HFDx is each Half Flux Diameter. Left chart is the case of the star disappeared and right chart is star appeared.

It will be considered how value is expected in various cases.

(1) Aperture includes background noise only. (The star is disappeared.)

The pale blue area is background which only has ccd's noise. If the noise is uniform in the background area, HFD is the diameter of circle which has area half of the aperture's area. It is calculated as..

If A is radius of measurement aperture. H is Half Flux Diameter then

(4)

(5)

As an example aperture radius is 7:

The radius of aperture is defined below.

A part(or entire) of a pixel that composes the measurement aperture is included in the circle which is drawn using the radius.

Therefore the actual radius(Rr) caluculated from the number of pixels is ..

(6)

N is number of pixels.

In the case of setting radius is 7 then N is 185. Then Rr=7.68 .

Hence, the diameter is 15.35 and HFD is estimated 10.86　.

If you want to estimate roughly, the radius can use 7.5 Or 7 .

(2) Aperture includes faint star.

When a faint star is appeared as the chart at right side of figure 2, the centre of star image forms a mountain and very faint skirts has extended around it. Therefore the HFD become smaller than the case of background only.

(3) Obvious star is displayed

It is the case of figure 1. The noise is least and the shape of star is stable. Then HFD is equal to FMHW.

(4) When there is bias in the distribution of the background noise.

Variance of Half Flux Diameter is increased. Especially, when the centre of aperture is on the valley of noise, the value of HFD will be a minus quantity. However, the mean of HFD will become to the value above mentioned.

4. HFD obtained from observation videos.

(1) Video has obvious star and least noise.

As an example, the observation video of the triple star XZ8808(SAO78233) and XZ86324.

Result of the analysis is shown in figure 3.

XZ86203 was disappeared at No.1016 frame, and the intensity of the star became to 95% at full brightness. And the light curve dropped to 50% at No.1410 frame. The change is not seen in HFD during the transition of luminous intensity. And, at No.1433 frame, HFD does not increase though there are only 6% intensity of light. Then, No.1434 frame when the star disappeared completely, the HFD increased to 13.5, and after this frame the mean value of HFD became about 11.

From this, we can understand that HFD reflects the appearance or disappearance of star sensitively.

Figure 3 Change of luminous intensity and Half Flux Diameter of

occultation SAO78233(XZ8808 double star) and XZ86324.

(2) The faint star case

Figure 4 shows the change of luminous intensity and HFD of the occultation lambda Aquarii in 1 Aug 2007 observed by Kunio Kenmotsu(Okayama, Japan). Kenmotsu had found a very faint star appeared 0.9 second before the appearance of main star. This is the graph drew from the .csv file sent from him to confirm the luminous increasing occurred by component actually. In this case, the “setting” radius of aperture is 7. Therefore the expected value of HFD of background is 10.68 . The graph shows that the frames before No.83 are scattered around 10. On the other hand, after No.84 the almost frame has value of HFD from 5 to 7.5 .

It is different from the case of background only, and it means thirty frames that has the concentrated brightness on the centre of aperture are continued. If the small increasing of brightness occurred by influence of earth shined lunar face then it is expected that the HFD becomes same as background (See (3) analyzing various place on the moon). From this, I concluded the light curve obtained from the observation is indicate a new component of double star.

Figure 4 Luminous change and Half Flux Diameter obtained from observation of the occultation of Lambda Aquarii.

(3) Half Flux Diameter of various places on the moon.

Various places on the moon are analyzed with HFD.

 Figure 5 Analyzed area Sky background (Indicated in yellow circle) There are a lot of noise because it was setting the gain high. Lunar Limb (Blue area) Lunar moves from the bottom of the image to top of the image. The lunar limb passes across the red circle(aperture) slowly. Therefore, if the site of aperture is fixed, then the measurement value increases slightly with lunar's movement. Top <- -> Bottom of image The brightness of pixel is changed at the moon limb. The left side is sky background and the right side is lunar's face. In this case the aperture is just measuring the boundary of both. Earth shined face (Magenta aera)

Table 1 includes Figure 5 shows the analyzed area. And Table 2 shows the result of HFD processing. It seems that the HFD of lunar limb and earth shined face is same as sky background.

 Sky background The radius of measurement aperture is 5. Then, the HFD value expected in the case of background is 7.68 . It seems the mean of HFD is about 7.5 . It well correspond to the theory. The lunar limb. The mean value is about 7.5 . The difference with background cannot be found. Earth shined lunar face. The diffecence with background cannot be found too.

5. Conclusion

Half Flux Diameter is convenient index because it can be processed fast and easy in computer. If there is no star profile then the value of HFD expected is obtained as (Radius of Aperture)/1.4 . The HFD of each frame has various value, and a mean value of them will become expected value above mentioned. When the star is appeared, HFD value is smaller than the case of no star. Using this, it can determine objectively the frame has star or not.

6. Ackowledgement

Kunio Kenmotsu sent me a .csv file of his observation. Masayuki Isida sent a video CD of grazing occultation of XZ4509. These data are useful for investigation about Half Flux Diameter and for improving Limovie. Many thanks for help of them.

7. Refference

Fast Auto-Focus Method and Software for CCD-based Telescopes [1]　By Larry Weber and Steve Brady

http://users.bsdwebsolutions.com/~larryweber/ITSPaper.htm