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

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) as the target. Some 930 single stars with good photometry and DR2 matchups, were used to calculate an initial fit, after which I redid the fit for the 99% of stars (922) which best matched the initial fit. I found that 2 colour fits, using BP−RP and G−RP, usually gave the best results, although the results seem to get less accurate towards the red end, for ultracool stars of M7 or redder.

The general equation is:
Target Colour  ≈  a0 + a1(BP−RP) + a2(BP−RP)² + a3(BP−RP)×(G−RP) + a4(G−RP) + a5(G−RP)²
The colour ranges are:
1.771 ≤ BP−RP ≤ 5.197, and 0.89 ≤ G−RP ≤ 1.68

Target Coloura0a1a2a3a4a5Residual Var.
V − G-0.12021(±3.3960)0.88909(±2.7323)0.0062748(±0.35493)0.059802(±3.0616)-0.998440(±9.7667)-0.060393(±5.9120)0.00151070
Rc − G-1.08640(±3.4010)0.62724(±2.7055)-0.0208210(±0.35844)0.033311(±3.0532)0.096642(±9.7295)-0.424650(±5.8772)0.00095219
G − Ic-0.81587(±3.4042)0.49131(±2.6980)0.0202830(±0.35146)-0.476880(±3.0176)1.659300(±9.7455)0.225040(±5.8636)0.00066358

For V−G, a simple polynomial using BP−G seems just as accurate at the one above, despite statements that BP is not very accurate for dim red stars:
V − G  ≈  −0.2084 + 0.92954×(BP−G) − 0.022026×(BP−G)²    (for 0.867 ≤ BP−G ≤ 3.61, Residual Variance : 0.0015067)

Gaia DR2 to 2MASS J H Ks

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 high proper motion stars (with V−Ks > 1.8 ) in the UCAC4 catalog as well as The Solar Neighborhood XXXV, as the target. Some 27148 stars (filtered to try and get single stars with good photometry) with DR2 matchups (BP−RP ≥ 0.8 ), were used to calculate an initial fit, after which I redid the fit for the 99% of stars (27144) 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 general equation for J is just like the V Rc Ic ones:
G − J  ≈  a0 + a1(BP−RP) + a2(BP−RP)² + a3(BP−RP)×(G−RP) + a4(G−RP) + a5(G−RP)²
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:
0.8 ≤ BP−RP ≤ 5.197, and 0.456 ≤ G−RP ≤ 1.897 (for J only)

Target Coloura0a1a2a3a4a5Residual Var.
G − J-0.083751(±0.15281)0.11537(±0.57344)-0.056927(±0.11011)0.50902(±0.80814)2.2595(±1.2913)-0.90936(±1.2276)0.00067525
Target Colourb0b1b2b3b4Residual Var.
G − H-1.2074(±0.25643)4.2167(±0.50855)-1.6933(±0.34551)0.36123(±0.096369)-0.028389(±0.0093147)0.0025329
G − Ks-1.1495(±0.25667)4.1566(±0.50903)-1.5619(±0.34589)0.31984(±0.096505)-0.024229(±0.0093339)0.0024061

Gaia DR2 to APASS B

Gaia does not provide a transformation to Johnson B, so to make my own, I matched Gaia DR2 magnitudes to a selection of UCAC4 stars that have B magnitudes from APASS (and Tycho-2). These are not necessarily all that accurate, but are better than using USNO B catalog magnitudes, at least for Red Dwarfs. The UCAC4 stars have V−Ks > 1.8, and are ‘high’ proper motion (more than 50mas/yr). I also added the restriction that BP−RP ≥ 0.8.

Some 24698 stars (filtered to try and get single stars with good photometry) with DR2 matchups , were used to calculate an initial fit, after which I redid the fit for the 99% of stars (23206) which best matched the initial fit. I found that a low-order polynomial seemed best (I did not try a linear fit) — multicolour fits or higher-order polynomials provided little improvement at best.

The general equation (Residual Variance is 0.019183) is :
B − G  ≈  −0.12204 + 1.3364×(BP−RP) − 0.094776×(BP−RP)²
The colour range is :
0.8 ≤ BP−RP ≤ 3.83

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 .

Target Coloura0a1a2a3a4a5Residual Var.
g′ − Rc2.3188(±0.7.2416)-1.9312(±0.10.32)0.97627(±4.421)-0.1294(±2.2253)0.054112(±2.2837)0.098408(±0.30229)0.0025772

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

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) because the photometry is real Rc and Ic, and goes to redder, dimmer stars than APASS will allow. Two color fits again proved to be the most accurate, although only marginally for V. The number of stars used is 855 (99% of 863).

The general equation is:
Target Colour  ≈  a0 + a1(UCAC−J) + a2(UCAC−J)² + a3(UCAC−J)×(J−Cx) + a4(J−Cx) + a5(J−Cx ,
where Cx is H for V − J  and Rc − J, and Ks for Ic − J.
The colour ranges are: 2.228 ≤ UCAC−J ≤ 6.745 ,  0.39 ≤ J−H ≤ 1.171 ,  and 0.643 ≤ J−Ks ≤ 1.324.

Target Coloura0a1a2a3a4a5Residual Var.
V − J2.3945(±3.1278)0.48740(±0.76665)0.015708(±0.048323)0.59453(±1.0492)-3.3444(±6.4798)0.78117(±3.2299)0.012624
Rc − J1.1711(±3.1090)0.57075(±0.76289)-0.016831(±0.047976)0.46052(±1.0396)-2.2027(±6.4374)0.30483(±3.2238)0.0074815
Ic − J1.2832(±3.9542)-0.17095(±0.66178)0.0038097(±0.057028)0.5399(±0.91717)-0.41556(±7.7583)-0.88529(±3.9461)0.0027074

For V−J, a simple polynomial using UCAC−J seems almost as accurate at the one above:
V − J  ≈  0.53715 + 0.91195×(UCAC−J) − 0.0071434×(UCAC−J)²    (for 2.228 ≤ UCAC−J ≤ 6.745, Residual Variance : 0.01297)

© John Q Metro, June 2018.