Proper motion and parallax of Wolf 359

Wolf 359, also known as CN Leo, is a red dwarf star of spectral type M6.5 [1] in the constellation of Leo near the ecliptic. With a distance of 7.9 lightyears, it is the eight-nearest star to our Sun. With a magnitude of 13.5, this star from our neighbourhood is only visible with a telescope. But remember, as a red dwarf, it only has about 0.00002 of the luminosity (visual) of our Sun [1].

Data on Wolf 359 from CDS/Aladin (2016) [2]:

$$ \begin{align} \alpha_{J2000} & = \hspace{3.5mm} 10:56:28.826208 \\ \delta_{J2000} & = +07:00:52.34400 \\ \mu_{\alpha} & = -3.80809 \ \text{arcsec/year} \\ \mu_{\delta} & = -2.69261 \ \text{arcsec/year} \\ \Pi & = \hspace{3.5mm} 0.41313 \ \text{arcsec} \end{align} $$

All images of Wolf 359 were taken with the Planewave CDK-14 (observatory K26) and a modified Canon 500d. Figure 1 has a total integration time of 152 minutes and is the result of the stacking of 76 individual images (exp. time: 120 s at ISO 1600). These images were taken on 5 different nights between February and April 2020 and show amongst others several galaxies and QSO's near Wolf 359 (field of view: 30 arcmin x 20 arcmin).

Wolf359_img
Figure 1: Wolf 359, a red dwarf star (at the centre of the image), only has 0.09 solar mass and 0.16 solar radius. This explains why this star, although close to our Sun, only has a magnitude of 13.5.
Instrumentation: Planewave CDK-14 and Canon 500d (obs.: K26); total: 76x2 min at ISO 1600 during 5 nights between February and April 2020. Processing software: Pixinsight.

Figure 1: full resolution (2177px·1452px; 0.22 MB)

Wolf359_img_annot
Figure 2 (identical to figure 1): Several galaxies and QSO's can be seen in this stretched and annotated image of Wolf 359 (field of view: 30'x20').

Figure 2: full resolution (2178px·1453px; 0.81 MB)

Wolf359_img_detail
Figure 3: Close-up of the field around Wolf 359 (February-April 2020).

Figure 3: full resolution (1200px·1200px; 0.38 MB)

Astrometry on Wolf 359

For the determination of the exact position of Wolf 359, the software Astrometrica [3] is used. Figure 4 shows an example from April 24, 2020. On average, 10 to 15 images per night of observation are stacked (images of bad quality are discarded). On the resulting image, 60 to 70 stars can be used by Astrometrica to determine the position of Wolf 359.

After several nights of observation, small deviations in the position of the target star can be observed. Beside the usual statistical deviations, another factor plays a role; particularly the fact that on each night a slightly different set of reference stars are used by Astrometrica. A higher accuracy can be obtained by checking the position of the star to a few reference stars next to the target (see also Berry [4]). The procedure adopted in the following is to use 4 reference stars and calculate their mean positions on the images taken on several nights. Next, the deviations of the 4 stars on each individual stacked image against their mean positions is calculated and applied to the target. Hence, a position of the target star with a higher accuracy can be obtained as can be seen in the next paragraph.

The following discussion includes observations from 18 nights between February 22, 2018 and April 26, 2020. Almost all observations have been done between the four months February and May of each year with one exception. One observation was done during October 2018; unfortunately under really bad circumstances. At that moment, the star was only visible 'through' the branches of a tree - needless to say that the quality of the resulting image is 'limited'.

Wolf359_astrometrica
Figure 4 Example of the position determination from April 22, 2020. Astrometrica uses on average 60 to 70 stars.

This image of April 22, 2020 is interesting as it shows the position of Wolf 356 as seen from the Earth. Not so particular you would say, but a day later, on April 23, 2020 the spacecraft New Horizons beyond the orbit of Pluto and Arrokoth took an image too of this star.

The parallax of a star is the effect of seeing a nearby star from different positions in space compared to distant background stars. Usually, this shift in position is due the the rotation of the Earth around the Sun. With a baseline corresponding to the orbit of the Earth, the parallax of Wolf 359 is 0.413 arcsec. Therefore, we have to wait for several months or half a year to detect this annual parallax.

In case of New Horizons, the effect of parallax is 'instantaneous', as both, the Earth and the spacecraft observe the same star at a specific date (April 23, 2020). As both observers are separated by 7 billion km, much more than the earthly baseline, the New Horizons parallax should be clearly visible in the images. More to come as soon as the images of New Horizons are available...

For more details, see the homepage of The New Horizons Parallax Program of the Johns Hopkins University and under #NHparallax [7].

Motion in Right Ascension (RA) and Declination (Dec) of Wolf 359

The following plots show the Right Ascension and the Declination versus (Julian) date. Without any proper motion or parallax, the star would have the same position at any date. The representation in both plots would be a horizontal line - but both plots look different.

Two different astronomical phenomena can be seen:

Both, RA and Dec versus date are fitted by the following function:

$$ y(t) = a \cdot t + b + c \cdot \sin \left( \frac{2 \pi}{365.25} \cdot t + d \right) $$

where a is the value of the proper motion μ.

Wolf359_RA Wolf359_RAerror
Figure 5 Top: RA versus date for Wolf 359 between February 2018 and April 2020. Error bars (0.18 arcsec) as given by Astrometrica.
Bottom: Difference between RA and the fitted function. The plotted error bars (0.05 arcsec) are the standard deviation of the differences between the measured positions and the fitted function.

Wolf359_Dec Wolf359_Decerror
Figure 6 Top: Dec versus date for Wolf 359 between February 2018 and April 2020. Error bars (0.15 arcsec) as given by Astrometrica.
Bottom: Difference between Dec and the fitted function. The plotted error bars (0.03 arcsec) are the standard deviation of the differences between the measured positions and the fitted function.

Astrometrica uses in the case of Wolf 359 60 to 70 stars for the position determination. Mean errors are 0.18 arcsec in RA and 0.15 arcsec in Dec. Figure 5 and 6 (bottom) give the differences between the measured position of the target and the fitted function. The standard deviation of these differences is clearly less then the errors given by Astrometrica; 0.05 arcsec in RA and 0.03 arcsec in Dec. Using a few reference stars (in this case 4 stars) close to the target star gives an improvement in the position determination.

Proper motion:

From both fitting functions (figure 5 and 6), the proper motion μ in RA and Dec can be calculated:

$$ \begin{align} \mu_{RA} & = -2.94633 \cdot 10^{-6} \ \text{deg/day} = -3.8741 \ \text{arcsec/year} \\ \mu_{Dec} & = -2.04971 \cdot 10^{-6} \ \text{deg/day} = -2.6952 \ \text{arcsec/year} \end{align} $$

Comparing with the data from CDS/Aladin (2016), the following relative errors can be calculated:

$$ \begin{align} \frac{\Delta \mu_{RA}}{\mu_{RA}} & = \frac{|-3.8081+3.8741|}{3.8081} = 1.73 \% \\ \frac{\Delta \mu_{Dec}}{\mu_{Dec}} & = \frac{|-2.6926+2.6952|}{2.6926} = 0.10 \% \end{align} $$

Some theory about the proper motion and parallax

Beside evaluating observational data, it would be interesting in making a theoretical description of the movement of a star across the sky including not only the proper motion but also the effect of the yearly parallax. For this purpose, the following equations can be applied:

$$ \begin{align} \alpha & = \alpha_{0} + \mu_{\alpha} \cdot n + \Pi \cdot P_{\alpha} \\ \delta & = \hspace{1mm} \delta_{0} + \hspace{0.5mm} \mu_{\delta} \cdot n + \Pi \cdot P_{\delta} \end{align} $$

Including the proper motion is not a difficult task; but finding and calculating the parallax factors Pα and Pδ requires more work. All you have to know is an initial position of the star α0 and δ0 (f.ex. J2000); the proper motions μα and μδ in RA and Dec and finally the parallax Π with the parallax factors. The following paragraphs show how to calculate them [5][6].

Number n of days since 2000, Jan 1 (J2000.0):

$$ n = JD - 2451545.0 $$

Mean longitude L of the Sun:

$$ L = 280.460^{\circ} + 0.9856474^{\circ} \cdot n $$

Mean anomaly g of the Sun:

$$ g = 357.528^{\circ} + 0.9856003^{\circ} \cdot n $$

Ecliptic longitude λ of the Sun (βSun = 0):

$$ \lambda = L + 1.915^{\circ} \cdot \sin g + 0.01997^{\circ} \cdot \sin 2g $$

Obliquity ε of the ecliptic:

$$ \varepsilon = 23.439^{\circ} - 0.0000004^{\circ} \cdot n $$

Distance Earth-Sun R:

$$ R = 1.00014 - 0.01671 \cdot \cos g - 0.00014 \cdot \cos 2g $$

Rectangular equatorial coordinates (X, Y, Z):

$$ \begin{align} X & = R \cdot \cos \lambda \\ Y & = R \cdot \cos \varepsilon \cdot \sin \lambda \\ Z & = R \cdot \sin \varepsilon \cdot \sin \lambda \end{align} $$

Parallax factors Pα and Pδ in right ascension and declination:

$$ \begin{align} P_{\alpha} & = -X \cdot \sin \alpha + Y \cdot \cos \alpha \\ P_{\delta} & = -X \cdot \cos \alpha \cdot \sin \delta - Y \cdot \sin \alpha \cdot \sin \delta + Z \cdot \cos \delta \end{align} $$

Knowing the initial position of the star (α0, δ0), the proper motion (μα, μδ) and the parallax factors (Pα, Pδ), Π being the parallax, the current position of the star can finally be calculated by the already mentioned equations:

$$ \begin{align} \alpha & = \alpha_{0} + \mu_{\alpha} \cdot n + \Pi \cdot P_{\alpha} \\ \delta & = \hspace{1mm} \delta_{0} + \hspace{0.5mm} \mu_{\delta} \cdot n + \Pi \cdot P_{\delta} \end{align} $$

Motion of Wolf 359 across the sky

Figure 7 shows the motion of Wolf 359 across the sky between February 2018 and April 2020. Error bars, calculated by Astrometrica, are given on the plot. The area represented in this plot corresponds to 12.6 arcsec x 9 arcsec. The theoretical trajectory has been fitted to the observational data with an offset of 0.72 arcsec in declination. As already mentioned above, the data point of October 2018 shows the largest deviation from the best fit. Overall, the main problem with all the fitting functions is that data are only obtained for the same 4 months of a year. As observational data for about 2/3 of the trajectory are missing (except for the data point of October 2018), a systematic error in the fitting curves can therefore not be excluded.

Wolf359_RA_Dec
Figure 7: Motion of Wolf 359 across the sky between February 2018 and April 2020. The theoretical trajectory is fitted to the observational data. The area corresponds to a field of 12.6 arcsec x 9 arcsec.

Parallax ellipse of Wolf 359

Finally, the effect of the parallax can be calculated by subtracting the initial position and the effect of proper motion from the observed positions. The resulting parallax ellipse is shown in figure 8 where we can see some special characteristics.

Wolf 359 is very close to the ecliptic. The apparent movement (parallax) of the star lies almost exactly on the plane of the ecliptic. Therefore, the parallax ellipse is seen from the side; it is almost a straight segment. Except for the October 2018 data point (not included in this plot), all the other observed positions are in acceptable agreement with the theoretical parallax ellipse with a value of 0.413 arcsec.

Wolf359_parallax_ellipse
Figure 8: Parallax ellipse for Wolf 359 from observations done between February 2018 and April 2020.

Some references