1. AD Canis Minoris: a δ Scuti Star in a Binary System
Roy Axelsen & Tim Napier-Munn
Astronomical Association of Queensland

2. δ Scuti Stars Short period pulsating variable stars
About 1 – 2.5 solar masses
Spectral classes A and F
δ Scuti stars studied by amateur astronomers have:
Periods of a few hours
Amplitudes of a few tenths of a magnitude in V
One period, or more than one period may be present
Periods can change gradually or suddenly

3. AD Canis Minoris
Variability first reported by Hoffmeister in 1934
Period: d (2 hr 57 min)
Amplitude: 0.3 V mag
Magnitude range: 9.25 – 9.55
Observations suggested a slowly increasing period (Jiang 1987; Rodriguez et al 1988, 1990; Yang et al 1992; Burchi et al 1993)

4. Rodriguez et al. IBVS 3427, 1990 (Fig. 1)
Rodriguez et al. IBVS 3427, 1990 (Fig. 1). O-C diagram showing residuals from a linear fit. A quadratic fit in the form:
O-C = a + bE + cE2 is therefore best

5. AD Cmi
O-C diagrams suggested combined quadratic and trigonometric functions provided the best fit (Fu & Jiang 1996; Fu 2000; Hurta et al 2007; Khokhuntod et al 2007)
Conclusion: AD CMi has a slowly increasing period, and the O-C diagram is modulated by the light travel time effect of a binary stellar system

6. Hurta et al 2007 (Fig. 1). O-C diagram of AD CMi with a combined quadratic and trigonometric function

7. Each point on an O-C diagram represents the difference (O-C) between the actual, observed time of a peak in the light curve, and the time the peak would have occurred if the period had remained constant during all observations

8. I was sceptical of these results because of the large variance here
Hurta et al 2007 (Fig. 1). O-C diagram of AD CMi with a combined quadratic and trigonometric function

11. AD CMi O-C Diagram Model: Where:
O - C = a + bE + cE2 + A sin ø + B cos ø
Where:
E is the epoch
a, b, c, A and B are parameters to be estimated from data
ø is the eccentric anomaly, the solution to Kepler’s equation:
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)

12. AD CMi O-C Diagram Kepler’s Equation: Where: Kepler’s equation:
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)
Where:
e is the eccentricity of the elliptical orbit
Porb is the orbital period of the binary system
Ppul is the pulsational period of the δ Scuti star
E is the epoch
T is the time of periastron of the assumed elliptical orbit
Kepler’s equation:
Must be solved iteratively for ø; it cannot be solved analytically

13. AD CMi O-C Diagram Kepler’s Equation:
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)
The right hand side of the equation is the mean anomaly, M, at a given time for a two body system
Thus:
M = 2 π (1/Porb) (Ppul E – T)
(We will need to return to M later)

14. AD CMi O-C Diagram Hurta et al 2007 Solution
O - C = a + bE + cE2 + A sin ø + B cos ø
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)

15. Hurta et al 2007 (Fig. 2). O-C diagram of AD Cmi after subtraction of the quadratic function. The resulting fit is entirely due to the light travel time effect of a binary system

17. AD Cmi: Orbital Period of the Binary System
Fu and Jiang 1996
30 y
Fu 2000
30.44 y
Hurta et al 2007
42.8 y
Khokhuntod et al 2007
27.2 y

18. AD CMi Photometry
DSLR photometry over 8 nights, January and February 2016
Telescopes:
2 nights – Celestron C9.25 SCT
6 nights – Orion ED80 refractor
Image capture:
C9.25: 800 ISO, 120 sec exposures, 15 or 20 sec gap
Orion ED80: 400 or 800 ISO, 120 or 225 sec exposures, 15 or 30 sec gap
Darks and flats taken

19. AD CMi Photometry
Photoelectric photometry over several nights in February 2011, but only one light curve peak was captured
Photometer:
SSP-5, Optec Inc, Lowell, Michigan, USA
Hamamatsu R6358 photomultiplier tube
Telescope:
Celestron C9.25 SCT

20. Data Analysis
PEP:
Non-transformed magnitudes were calculated from counts through a photometric V filter
The peak of the light curve was taken as the time of the maximum value of a 5th order polynomial expression fitted to the light curve in Microsoft Excel
The time of maximum (in HJD) was determined by interpolation on visual inspection of the light curve

21. AD CMi PEP Light Curve

22. Data Analysis DSLR Photometry:
Aperture photometry performed in AIP4Win
Transformed magnitudes in V calculated
Transformation coefficients previously determined from images of standard stars in the E regions
The peak of each light curve was taken as the time of the maximum value (in HJD) of a 10th order polynomial function fitted to the light curve in PERANSO

23. AD CMi DSLR Photometry Light Curve

24. Results 118 times of maximum (TOM) tabulated
73 from the most recent publication (Khokhuntod et al 2007)
36 added from subsequent literature and the AAVSO international database
9 from personal data

25. AD CMi O-C Diagram

26. AD CMi O-C Diagram Axelsen & Napier-Munn Solution
O - C = a + bE + cE2 + A sin ø + B cos ø
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)
That is:
M = 2 π (1/Porb) (Ppul E – T)
Software to derive the solution to the above using only O-C values and epochs could not be found
Therefore we decided to “reverse engineer” the model using e, Porb, and T from the paper of Hurta et al 2007, and Ppul from a least squares linear ephemeris calculated from all recorded epochs

27. AD CMi O-C Diagram Axelsen & Napier-Munn Solution
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)
That is:
M = 2 π (1/Porb) (Ppul E – T)
As noted, the solution for ø must be found iteratively
We used a VB macro routine in Microsoft Excel provided by Burnett (1998)
This requires three arguments: the mean anomaly M, eccentricity e and a convergence criterion
M is easily calculated, given Porb, Ppul, and T, for a given epoch, E
For each O–C, E, and φ data point, the model was then fitted by non-linear regression using the Minitab 16 statistical software package (Minitab 2016).

28. AD CMi O-C Diagram Axelsen and Napier-Munn 2016 Solution
O - C = a + bE + cE2 + A sin ø + B cos ø
ø – e sin ø = 2 π (1/Porb) (Ppul E – T)

29. AD CMi O-C Diagram O - C = a + bE + cE2 + A sin ø + B cos ø

30. AD CMi O-C Diagram O - C = A sin ø + B cos ø

31. AD CMi O-C Diagram O - C = a + bE + cE2

33. Hurta et al 2007 (Fig. 1). O-C diagram of AD CMi with a combined quadratic and trigonometric function

35. Rate of Increase in the Pulsational Period of AD CMi
Authors
Rate of Period Change
(Error) d yr-1
Fu & Jiang 1996
0.14 x 10-8
Fu 2000
0.10 x 10-8
Hurta et al 2007
1.15 (0.01) x 10-8
Khokhuntod et al 2007
0.26 (0.02) x 10-8
Axelsen & Napier-Munn 2016
0.61 (0.07) x 10-8
After about 80,000 periods or about 28 years, a peak in the light curve will occur 10 minutes later than it would have if the period had remained unchanged (using results of Axelsen and Napier-Munn)

36. Conclusions
A literature review, extraction of data from the AAVSO International Database, and personal PEP and DSLR photometric data have substantially extended the O-C data on AD CMi
The published dataset now comprises 118 times of maximum, 1959 to 2016
The most recent previously published dataset comprised 81 times of maximum (Hurta et al 2007)
Nine times of maximum are from personal observations

37. Conclusions
It is confirmed that the O-C data is best modelled by a combined quadratic and trigonometric function O - C = a + bE + cE2 + A sin ø + B cos ø, indicating that the O-C data is modulated by the light time effect of a binary stellar system
The current model improves on that of Hurta et al (2007)
The quadratic part suggests that the pulsational period of AD CMi is increasing at a constant rate (0.07) x 10-8 d yr-1

38. Conclusions
Because of considerable variation in the published orbital period of the system, and in the published rate of decrease in the pulsational period, it may be many years before these values can be established accurately from photometric studies