Photometric conversions in Guides 7.0 and 8.0
Last updated 23 May 2002
There have been a few inquiries as to how Guide handles problems such as converting Tycho magnitudes to Johnson, and vice versa. Those questions will doubtless only increase with Guide's new capability to show stars in a user-selected magnitude band.
Guide does two sorts of magnitude system conversions. Using Tycho-2 VT and BT data, it computes Johnson B, V, and U, and Cousins R and I data. Using USNO-Ax.0 R and B data, it computes Johnson V magnitudes. So far, at least, that's all the converting it is able to do.
There are three main conversion problems. The biggest is converting Tycho-2 VT and BT to Johnson V and B. (Concerning the other two main conversion problems, you can click here for information on transforming VT and BT to R, I, and U, and you can click here for comments on converting Ax.0 B and R magnitudes to V.)
There are no less than three different ways of converting VT/BT data to approximate Johnson data. In ascending order of complexity, they are:
The "easy" way: On page 57 of the Introduction and Guide to the Data that accompanies the Tycho/Hipparcos CDs, we see the following easy-to-use formulae:
Vj = VT - 0.09 (BT-VT) (B-V)j = .85 * (BT-VT)
...that is, it's assumed that VT and BT are linearly related to their Johnson counterparts. Both formulae are good only over the range -.2 < (B-V)t < 1.8.
It also says that the above gives "errors generally below 0.015 mag in Vj and below .05 mag in (B-V)j. The transformations apply to unreddened stars and ignore variations due to luminosity class. They should not be applied to M-type stars, even for (B-V)t < 1.8." Guide doesn't support this "easy" transformation method at all.
The "medium" way: On the same page, a slightly better way is offered. The difference between the values given by the above formulae, and values given by a more accurate analysis, are given for values of (B-V)T of -.2, .1, .5, 1.4, and 1.8. Between those values, you have to do a linear interpolation. Thus, the relationship is still a linear one, but is broken in several places... a "piecewise linear" relationship. The resulting (B-V)j and Vj data have "deviations less than .005 mag from a more accurate spline fit." This is the method currently use in Guide 7.0 and, by default, in Guide 8.0.
The "hard" way: Previously, the above "medium" way was the the most accurate method I'd heard of. However, Brian Skiff suggested a better method on the Minor Planet Mailing List. He posted a data table and the following note:
...To get standard V and B from Tycho-2, it is probably best to use the relation shown by Mike Bessell in the July 2000 PASP... Bessell does not give an algebraic relation, but instead shows a cubic spline fit with a look-up table... I have copied out Bessell's table below as a flat ASCII list.
Bessell's table gives, for (B-V)T values from -.25 to 2.0, values from which Johnson magnitudes (B and V) and Hp (Hipparcos magnitudes) can be computed. I wrote a little piece of code to do cubic interpolation within this table. The table, and the code used for the interpolation, is available on my Web site:
The table and source code are given in the file COLORS2.CPP. (COLORS.CPP, also available on that page, is the code for the "medium way".) I am assuming, and am fairly certain, that the newer transformation given by Bessell via Skiff and used in COLORS2.CPP makes the older "medium way" obsolete.
By default, Guide 8 makes use of the "medium" method. This is fast and easy to compute, and switching to the "hard" Bessell method is of dubious value; almost always, the difference in magnitude is far less than the errors in the magnitudes. However, some people have expressed an interest in having Guide use the Bessell method. If you're among them, be sure to download the current Guide software, because Guide 8 as distributed on the CDs lacked any knowledge of the Bessell method. Then edit the file win_menu.dat in Notepad or a similar text editor, and look for this line:
# 2075 748 Bessell magnitudes
Change the '#' to a space, and fire up Guide. Under the Extras menu, you'll see a new "Bessell magnitudes" option. Turn this on, and the menu entry will be checked and you'll see a slight change in Johnson V and B-V data for Tycho stars, when you click on them; when they're labelled with Johnson V mags; and in the "more info" section. The difference is very slight, usually under .01 mags.
Converting Tycho magnitudes to R, I, and U: Any method for doing this is apt to be shaky at best. But I know of two different ways of going about it.
I learned of the first (and simplest) method via an e-mail from Bruce Sumner:
An unpublished independent linear transformation set has been derived by Arne Henden, using Landolt standard equatorial stars (about 270 stars, V brighter than 10.5, B-V bluer than 1.8) and Tycho-1 magnitudes and colours: Uj = BT - 0.325 + 0.8034*(B-V)T Bj = BT + 0.018 - 0.2580*(B-V)T Vj = VT + 0.008 - 0.0988*(B-V)T Rc = VT - 0.014 - 0.5405*(B-V)T Ic = VT - 0.039 - 0.9376*(B-V)T Henden comments that U and I have terrible systematics, but that the BVR fits are quite good. Additionally no discrimination was made on luminosity class, and in all cases the relationships were assumed to be purely linear.
(Note that the above provides a fourth way of converting BT and VT to Vj and Bj... basically, the "easy" way with slightly different coefficients.) In Guide 8.0, I use the above formulae in computing Uj, Rc, and Ic from BT and VT.
The second (and more difficult, but possibly more accurate) way of transforming these colors would be to use the Bessell method, inverted, to compute Vj and Bj from BT and VT. Brian Skiff provided me with formulae to compute V-R as a polynomial:
delta = (B-V)j - 1 V-R = c0 + c1 * delta + c2 * delta^2 + ... + c11 * delta^11 with c0 = -0.4140951, c1 = 0.7357165, c2 = -0.05242979, c3 = -0.6293304, ...
This is claimed to be valid for -.23 < (B-V)j < 1.95. Brian Skiff also provided similar series to compute B-V from V-R, and V-R and B-V from V-I. These can be inverted with a root-finding routine. See COLORS.CPP for a C implementation of these polynomials and the 'inversion' routine.
So far, it seems to me that any conversion of B-V to other colors is apt to have extremely rough results that nobody should take seriously. (Conversion of V-I or V-R to B-V appears to be much more accurate.) I've used Arne Henden's simpler formulae, on the assumption that the added horsepower of the Skiff formulae may be a case of "measure with micrometer, cut with chain saw".
Computing V from USNO-Ax.0 B and R magnitudes: Conversion of Ax.0 magnitudes is a somewhat pointless task, since the photometric calibration is (in the words of its author, Dave Monet) "about as poor as one can have and still claim that the magnitudes mean something." However, I've adopted the following simple formula from T. Kato:
V = .375 * B + .625 * R = (3*B + 5*R) / 8
B-V = .625(B-R)
Given that the Ax.0 magnitudes are terrible to begin with, there didn't seem to be any point in adopting a more precise transformation.