Photometric Transforms for K and M Dwarfs

The photometric transforms I've come up with here were created as part of my work with the NStars program and my list of nearby stars. Quite a lot of stars have little in the way of standard Johnson-Cousins photometry available, and quite often one will only see vague guesses for V, or poorly calibrated plate magnitudes.

When it comes other types of photometry, however, more is available. I am referring less to the established alternate photometric systems, than I am to custom bandpasses and systems used by the likes of the US Navy UCAC and URAT, the MEarth project, or the ESA's Gaia. Transformations to Johnson-Cousins from these bandpasses are either not found in the literature, or otherwise inadequate.

In the end, I've recently ended up deriving my own, using typical Polynomial and Multilinear fit methods. I am not a professional Astronomer or Statistician, so I won't make any guarantees about the accuracy of these fits, but they seem to be reasonable to me.

• Gaia DR2 to Johnson-Cousins V Rc Ic
• Gaia DR2 to 2MASS J H Ks
• Gaia DR2 to Johnson B
• APASS g′ r′ i′ to Cousins Rc Ic
• UCAC and 2MASS to Johnson-Cousins

Gaia DR2 to Johnson-Cousins V Rc Ic

Updated October 10, 2018.

While the Gaia DR2 teams provide transforms from BP−RP to V R I, I find these unsatisfactory, mostly because they are limited to BP−RP < 2.75, which does not cover most Red Dwarfs. So I've derived my own fits.

The fits below use V Rc Ic photometry from The Solar Neighborhood XXXV (Winters+ 2015), The SolarNeighborhood XXXVIII (Winters+ 2016), and CCD Parallaxes for 309 Late-type Dwarfs and Subdwarfs (Dahn+ 2017) (V and Ic) as the target. Around 934 to 991 stars with good photometry and DR2 matchups, were used to calculate the initial fits, after which I redid the fits for the 99.5% of stars which best matched the initial fits. I found that 2 colour fits, using BP−RP and G−RP, usually gave the best results, although only marginally for V. It should be noted that only a few of the stars were 'Ultracool Dwarfs'.

The general equation is:
Target Colour  ≈  a₀ + a₁(BP−RP) + a₂(BP−RP)×(G−RP) + a₃(G−RP)
The colour ranges are:
1.771 ≤ BP−RP ≤ 5.197, and 0.88 ≤ G−RP ≤ 1.59

For V−G, a linear fit using BP−G is pretty good as well:
V − G  ≈  −0.26726 + 1.0042×(BP−G)    (for 0.814 ≤ BP−G ≤ 3.61, Standard Error: 0.0369)

In many cases, the Gaia DR2 BP magnitude is unreliable (mag error >= 0.025), due to the objects being too dim and red, or because of contamination from a nearby hotter star. So I have also derived some transformations that use just G−RP.  959 to 1032 stars were used (the exact counts differ for each magnitude) with G−RP ≥ 0.88. It should be noted that scatter increases the redder you go, especially G−RP ≥ 1.5, so the results are not very precise for Ultracool Dwarfs, especially for V.

The polynomial equation for 0.89 ≤ G−RP ≤ 1.65   (99.5% of the stars is this range were kept) is:
Target Colour  ≈  a₀ + a₁(G−RP) + a₂(G−RP)² + a₃(G−RP)³

Gaia DR2 to 2MASS J H Ks

Updated October 7, 2018.

As with V R I, the Gaia teams provide transformations, and these transforms do go deep into the red. However, implementing and spot checking these transforms showed to me that the results, at least for brighter Red Dwarfs, are systematically off, with J being a bit too bright, H being a bit too dim, and Ks being too bright to a larger degree. The Ks results in particular prompted me to try my own fits.

The fits below use J H Ks photometry from 2MASS, using a selection matched to stars within 150 LY (with G > 6 ) in the Gaia DR2 catalog, filtered for good quality photometry, parallaxes, and having no bright stars nearby (White Dwarves are also excluded). Some 18827 Stars and Brown Dwarfs (BP−RP ≥ 0.488 ), were used to calculate an initial fit, after which I redid the fit for the 99.5% of stars (18734) which best matched the initial fit. I found that a 2 colour fit using BP−RP and G−RP gave the best results for G−J, but polynomials gave the best results for H and Ks.

The equation for J is fairly simple (the Standard Error is is 0.031):
G − J  ≈  −0.0070518 + 0.37295×(BP−RP) + 1.5529×(G−RP)

For H and Ks it is:
Target Colour  ≈  b₀ + b₁(BP−RP) + b₂(BP−RP)² + b₃(BP−RP)³ + b₄(BP−RP)⁴
The colour ranges are:
0.49 ≤ BP−RP ≤ 4.9, and 0.29 ≤ G−RP ≤ 1.56 (for J only)

Also, in many cases, the Gaia DR2 BP magnitude is unreliable, due to the objects being too dim and red, or because of contamination from a nearby hotter star. So I have also derived some transformations that use just G−RP. Over 20600 stars were used (the exact counts differ for each magnitude) with G−RP ≥ 0.3, but I found it best to use 2 (overlapping) ranges for each, since the color-color plots showed a turn with increased scatter when G−RP ≥ 1.5.

The polynomial equation for 0.3 ≤ G−RP ≤ 1.55   (99.5% of the stars is this range were kept) is:
Target Colour  ≈  a₀ + a₁(G−RP) + a₂(G−RP)² + a₃(G−RP)³

The linear equations for for 01.47 ≤ G−RP ≤ 1.79   (99% of the stars is this range were kept) are:

G − J    ≈  −3.7121 + 5.0946×(G−RP)   (Standard Err. : 0.127)
G − H   ≈  −4.4121 + 5.9605×(G−RP)   (Standard Err. : 0.184)
G − Ks  ≈  −5.1153 + 6.6719×(G−RP)   (Standard Err. : 0.221)

Gaia DR2 to Johnson B

Updated October 7, 2018. Take 2.

Gaia does not provide a transformation to Johnson B, so to make my own, I matched Gaia DR2 stars with good parallaxes (and 2MASS cross-matches) within 150 LY to APASS and Tycho-2 stars. Johnson B was calculated (using the Mamajek+ 2012 formulas) from Tycho-2, and used in preference to APASS-B for B < 11.5. I also calculated B from APASS g’ r’ i’, and used that in preference to APASS-B if APASS-B was more than 0.1 mag brighter than the calculated-B, in order to try and get past the reported ‘Red Leak’.

Some 12454 stars (filtered to discard anomalously low B−G values, and also White Dwarfs) were used to calculate an initial fit, after which I redid the fit for the 99% of stars (12306) which best matched the initial fit. I found that a low-order polynomial seemed best — multicolour fits or higher-order polynomials provided little improvement at best.

The equation (Standard Error is 0.084) is :
B − G  ≈  −0.39137 + 1.6034×(BP−RP) − 0.15377×(BP−RP)²
The colour range is :
0.49 ≤ BP−RP ≤ 3.9
The results seem reasonable for most stars, but for Red Dwarfs seem to be (still) too bright.

APASS g′ r′ i′ to Cousins Rc Ic

The APASS survey, in addition to Johnson B and V, also provide ‘primed’ SDSS magnitudes — g′ r′ i′. You can transform these to Johnson-Cousins using existing transforms. I have been using these equations and then Lupton 2005, but I ended up noticing that the computed values for Rc for Red Dwarfs were too dim, with the difference getting larger the redder the star.

So, using the APASS photometry in the UCAC4 catalog, I matched them up with the stars in The Solar Neighborhood XXXV. Some 603 single stars with good photometry and APASS magnitudes were used for the initial fit, the 99% (597) that best matched the initial fit were used to come up with the fit below.

The general equation for Rc is a two-colour fit:
g′ − Rc  ≈  a₀ + a₁(g′−r′) + a₂(g′−r′)² + a₃(g′−r′)×(g′-i′) + a₄(g′-i′) + a₅(g′-i′)²
The colour ranges are:
1.029 ≤ g′−r′ ≤ 1.631, and 1.628 ≤ g′-i′ ≤ 4.418 .

I've also derived a transform to Ic (Residual Variance 0.0050653), although it does not seem to be necessary, since the equations from the SDSS website seem to be good enough:
g′ − Ic  ≈  0.085531 + 1.2441×(g′-i′) − 0.01608×(g′-i′)²
The colour range is :
1.034 ≤ g′−i′ ≤ 4.418

UCAC and 2MASS to Johnson-Cousins

Updated October 12, 2018.

The US Naval Observatory started an independent all-sky survey in 1998, known as the USNO CCD Astrograph Catalog (UCAC). A single bandpass, 579-642 nm (between V and R ), was used (Simbad rather naughtily treats this as R, but the magnitudes are not close to Rc). The fourth revision, UCAC4 (2012), contains the best photometry, combining calibrated UCAC, APASS, Tycho-2, and 2MASS magnitudes in 1 catalog (UCAC5 merely cross references UCAC with Gaia DR1, and is missing the APASS and Tycho-2 photometry).

Even with APASS, many Red Dwarfs are lacking in measured Johnson-Cousins magnitudes. As an optical bandpass, UCAC magnitudes can be used in combination with 2MASS to provide better estimates for B V Rc Ic than 2MASS alone, and the cross reference with APASS provides a ready source of target magnitudes.

With that in mind, I'll still start with using The Solar Neighborhood XXXV (Winters+ 2015), The SolarNeighborhood XXXVIII (Winters+ 2016), and CCD Parallaxes for 309 Late-type Dwarfs and Subdwarfs (Dahn+ 2017) because the photometry includes real Rc and Ic, and goes to redder, dimmer stars than APASS will allow. Also, the color-color charts using APASS V show some strange features. Two color fits involving 2MASS were only marginally better than linear or polynomial fits, so I won't provide them. Around 800 stars (the exact amount depends on the color) with UCAC4 cross-matches were used, some outliers were discarded for each, with my usual 99.5% kept for the given fits.

The equation for V (the Standard Error is 0.095):
V − J  ≈  0.45979 + 0.95995×(UCAC−J)

The equation for Rc (the Standard Error is 0.077):
Rc − J  ≈  -0.21441 + 0.95995×(UCAC−J) − 0.029608×(UCAC-J)²

The equation for Ic (the Standard Error is 0.044):
Ic − J  ≈  0.22224 + 0.32142×(UCAC−J)

The colour range is: 2.228 ≤ UCAC−J ≤ 6.745, although only a few stars had UCAC−J > 5.3

Also, if G is available, you can use a 2 color fit for V, which more accurate than using UCAC−J or G−J alone (Standard Error 0.054):
V − J  ≈ 0.0684 − 0.030237×(UCAC−J) + 0.094171×(UCAC−J)×(G−J) + 1.0978×(G−J)
The colour ranges for G−J is 1.937 ≤ G−J ≤ 3.992.

© John Q Metro, June to October 2018.