# The Mass-Temperature Relation for B and Early A Stars Based on IUE Spectra
of Detached Eclipsing Binaries

Nancy Remage Evans Center for Astrophysics $|$ Harvard & Smithsonian
60 Garden St.
Cambridge, MA 02138, USA  Mckenzie G. Ferrari University of Massachusetts-
Dartmouth
285 Old Westport Rd.
North Dartmouth, MA 02747  Joanna Kuraszkiewicz Center for Astrophysics $|$
Harvard & Smithsonian
60 Garden St.
Cambridge, MA 02138, USA  Steven Silverberg  Center for Astrophysics $|$
Harvard & Smithsonian
60 Garden St.
Cambridge, MA 02138, USA  Joy Nichols  Center for Astrophysics $|$ Harvard &
Smithsonian
60 Garden St.
Cambridge, MA 02138, USA  Guillermo Torres  Center for Astrophysics $|$
Harvard & Smithsonian
60 Garden St.
Cambridge, MA 02138, USA  Makenzi Fischbach  Wellesley College
Wellesley, MA

###### Abstract

Ultraviolet spectra were taken of 25 Detached Eclipsing Binaries (DEBs) with
spectral types O, B, and early A with the International Ultraviolet Explorer
(IUE) satellite in the 1150 to 1900 Å region. The spectra were compared with
BOSZ model atmospheres (Bohlin, et al. 2017). The composite spectra of the
DEBs were modeled by a combination of models representing the hot and cool
components, and the temperatures of the hottest components of the systems were
determined. From these temperatures a direct Mass-Temperature relation was
obtained for stars close to the main sequence with solar metallicity for B and
early A stars:

log $M_{\odot}$ = -5.90 $\pm$ 0.27 + (1.56 $\pm$ 0.07) x log T

This relation allows a mass to be inferred for comparable stars from an
ultraviolet spectrum. The five chemically peculiar Am stars in the sample have
larger radii than normal A stars of the same mass.

early-type stars — ultraviolet spectroscopy — eclipsing binary stars —
effective temperature

##  1 Introduction

Observationally measured masses are critical to understanding stars. Evolution
calculations are tied to these measures. Eclipsing binary systems where the
inclination of the system can be determined are a crucial part of the sample
of stars for which we have accurate masses. Andersen (1991) compiled a list of
detached eclipsing binaries (DEBs) with accurately determined masses. This
list has subsequently been updated by Torres, Andersen, and Gimenez (2010; TAG
below).

Well covered photometric and spectroscopic curves analyzed with modern
software provide highly accurate masses, radii, and temperature ratios between
the two components of the system. However a valuable parameter in relating
these properties to, for example, evolutionary tracks is the temperature of
each of the two stars. These must be determined from, for instance, spectral
types.

When the Andersen list was drawn up, the International Ultraviolet Explorer
(IUE) Satellite was operating. It provided well calibrated spectra
particularly in the 1150 to 1900 Å ultraviolet (UV) region. For stars of A
spectral types and hotter, a large fraction of the flux is contained in this
spectral region. For this reason a program was set up to observe eclipsing
systems from the Andersen list with primaries in this spectral type range.
Comparison of these spectra with atmospheric models provides a temperature for
the primary. The goal of this study is to obtain a temperature which describes
the energy distribution in this important wavelength region. This results in a
direct mass-temperature relation for hot stars, which can be used to infer the
mass of a star from ultraviolet spectra.

##  2 Observations

The International Ultraviolet Explorer (IUE) Satellite was launched in 1978
and was operated by NASA, ESA and the British SERC. It provided spectra in the
wavelength regions 1150 to 1975 Å and 1910 to 3300 Å, in two resolution modes
(resolution 0.1 and 6 Å respectively for high and low resolution). Full
details of the satellite, spectrographs and cameras are given in Harris and
Sonneborn (1987).

For eclipsing binaries which have an O, B, or early A component we obtained
low resolution spectra with the IUE satellite in the short wavelength region
through the large aperture. These were largely from the program TEPNE (PI:
Evans), supplemented by archival data when available. The observations are
summarized in Table 1. Columns list the variable star designation, the HD or
BD number, the exposure time, the spectrum number, the observing date and the
observing time, the maximum data count (Data Numbers) in the spectrum and the
background count. Most spectra were well exposed; however a maximum count of
255 means some part of the spectrum is saturated and cannot be used.

Table 1: IUE Satellite Spectra for DEBs  System | ID | Exposure (s) | Spectrum ID | Obs Date | Obs Time | Max Count | Background
---|---|---|---|---|---|---|---
V478 Cyg | HD 193611 | 427.5 | SWP48207 | 1993/07/22 | 16:50:08 | 72 | 16
AH Cep | HD 216014 | 89.6 | SWP10148 | 1980/09/16 | 5:07:28 | 255 | 17
V578 Mon | HD 259135 | 2699.5 | SWP17905 | 1982/09/10 | 9:21:56 | 178 | 58
QX Car | HD 86118 | 4.8 | SWP50174 | 1994/03/06 | 22:20:14 | 179 | 14
V539 Ara | HD 161783 | 2.7 | SWP48476 | 1993/08/27 | 16:25:07 | 132 | 16
CV Vel | HD 77464 | 4.4 | SWP50160 | 1994/03/03 | 20:46:04 | 118 | 16
U Oph | HD 156247 | 10.5 | SWP48498 | 1993/08/29 | 15:14:27 | 189 | 15
V760 Sco | HD 147683 | 41.6 | SWP48497 | 1993/08/29 | 14:10:13 | 138 | 15
GG Lup | HD 135876 | 8.5 | SWP48496 | 1993/08/29 | 12:58:18 | 255 | 17
$\zeta$ Phe | HD 6882 | 1.1 | SWP48495 | 1993/08/29 | 11:37:37 | 143 | 16
$\chi^{2}$ Hya | HD 96314 | 6.8 | SWP50685 | 1994/05/03 | 20:28:43 | 108 | 15
IQ Per | HD 24909 | 111.7 | SWP50177 | 1994/03/07 | 2:19:11 | 192 | 16
PV Cas | HD 240208 | 880.5 | SWP48206 | 1993/07/22 | 15:43:50 | 99 | 19
V451 Oph | HD 170470 | 208.8 | SWP48217 | 1993/07/23 | 18:52:14 | 181 | 16
WX Cep | HD 213631 | 2999.1 | SWP50179 | 1994/03/07 | 20:38:38 | 109 | 19
TZ Men | HD 39780 | 29.8 | SWP50175 | 1994/03/06 | 23:26:55 | 164 | 15
V1031 Ori | HD 38735 | 69.5 | SWP50176 | 1994/03/07 | 0:39:52 | 132 | 15
$\beta$ Aur | HD 40183 | 0.7 | SWP50172 | 1994/03/06 | 19:34:58 | 114 | 14
YZ Cas | HD 4161 | 39.6 | SWP26934 | 1985/10/14 | 14:00:51 | 183 | 18
V624 Her | HD 161321 | 85.5 | SWP48216 | 1993/07/23 | 17:35:04 | 104 | 16
GZ CMa | HD 56429 | 419.7 | SWP50173 | 1994/03/06 | 20:55:47 | 126 | 16
V1647 Sgr | HD 163708 | 99.8 | SWP48475 | 1993/08/27 | 15:35:53 | 150 | 15
EE Peg | HD 206155 | 119.5 | SWP50684 | 1994/05/03 | 19:04:41 | 139 | 17
VV Pyx | HD 71581 | 77.7 | SWP50161 | 1994/03/03 | 22:08:02 | 189 | 15
AY Cam | BD+77∘ 328 | 2699.5 | SWP29742 | 1986/11/24 | 1:41:45 | 63 | 21

Table 2 provides the orbital phases at which the IUE observations were
obtained. Columns list the system, the period, the HJD of primary minimum, the
reference for the orbit, the phase of secondary minimum, the HJD of the IUE
observation, the orbital phase of the IUE observation, and notes. Most of the
IUE observation were taken outside of eclipse. In the final column asterisks
indicate 4 systems for which the observation occurred during some part of an
eclipse. Three of these systems (V478 Cyg, AH Cep, and V587 Mon) were
discarded during spectral fitting, as discussed in Section 5. V539 Ara was
also mildly affected by an eclipse, as is also discussed in Section 5.

Table 2: Phases of Observation

System | P | Min I | Ref | Phase of Min II | IUE |  IUE Phase | Notes
---|---|---|---|---|---|---|---
| (d) | HJD |  |  | HJD |  |
V478 Cyg | 2.88090063 | 2444777.4852 | 1 | 0.5067 | 2449191.2014 | 0.0613 | *
AH Cep | 1.774761 | 2445962.7359 | 2 | 0.5000 | 2444498.7135 | 0.0876 | *
V578 Mon | 2.40848 | 2449360.625 | 3 | 0.4507 | 2445222.8902 | 0.0141 | *
QX Car | 4.4780399 | 2443343.41512 | 4 | 0.4010 | 2449418.4307 | 0.6238 |
V539 Ara | 3.169112 | 2445056.777 | 5 | 0.3307 | 2449227.1841 | 0.9545 | *
CV Vel | 6.889494 | 2442048.66894 | 6 | 0.5000 | 2449415.3653 | 0.2652 |
U Oph | 1.6773458 | 2442621.6212 | 7 | 0.5017 | 2449229.1350 | 0.2675 |
V760 Sco | 1.7309295 | 2443250.8268 | 8 | 0.5101 | 2449229.0904 | 0.7881 |
GG Lup | 1.84960 | 2446136.7398 | 9 | 0.5064 | 2449229.0404 | 0.8754 |
$\zeta$ Phe | 1.6697724 | 2441643.6890 | 10 | 0.5000 | 2449228.9844 | 0.7122 |
$\chi^{2}$Hya | 2.267701 | 2442848.6107 | 11 | 0.5000 | 2449476.3532 | 0.6704 |
IQ Per | 1.74356214 | 2444290.3664 | 12 | 0.5222 | 2449418.5966 | 0.2374 |
PV Cas | 1.75047 | 2449210.7956 | 13 | 0.5027 | 2449191.1554 | 0.7801 |
V451 Oph | 2.19659557 | 2445886.53335 | 14 | 0.4978 | 2449192.2862 | 0.9438 |
WX Cep | 3.3784535 | 2425088.537 | 15 | 0.5000 | 2449419.3601 | 0.7635 |
TZ Men | 8.56900 | 2442403.7085 | 16 | 0.5096 | 2449418.4770 | 0.6216 |
V1031 Ori | 3.405565 | 2444643.665 | 17 | 0.5000 | 2449418.5276 | 0.0765 |
$\beta$ Aur | 3.96004673 | 2452827.195693 | 18 | 0.5000 | 2449418.3159 | 0.1819 |
YZ Cas | 4.46722236 | 2445583.78664 | 19 | 0.5000 | 2446353.0839 | 0.2093 |
V624 Her | 3.894977 | 2440321.005 | 20 | 0.5000 | 2449192.2326 | 0.6072 |
GZ CMa | 4.8008500 | 2443581.56132 | 21 | 0.5000 | 2449418.3720 | 0.7869 |
V1647 Sgr | 3.28279251 | 2441829.69510 | 22 | 0.2621 | 2449227.1499 | 0.4031 |
EE Peg | 2.62821423 | 2440286.4349 | 23 | 0.5000 | 2449476.2949 | 0.6176 |
VV Pyx | 4.5961832 | 2444620.65895 | 24 | 0.4804 | 2449415.4222 | 0.2054 |
AY Cam | 2.73496794 | 2443572.74441 | 25 | 0.5000 | 2446758.5706 | 0.8496 |

Note. — *Starred systems are where IUE observations overlap eclipse;
References: 1. Wolf et al. 2006; 2. Holmgren et al. 1990; 3. Hensberge et al.
2000; 4\. Andersen et al. 1983; 5. Clausen 1996; 6. Yakut et al. 2007; 7. Vaz
et al. 2007; 8\. Andersen et al. 1985; 9. Andersen et al. 1993; 10. Andersen
et al. 1983; 11. Clausen and Nordstrom 1978; 12. Wolf et al. 2006; 13.
Barembaum & Etzel 1995; 14. Clausen et al. 1986; 15\. Popper 1987; 16.
Andersen et al. 1987; 17. Andersen et al. 1990; 18. Southworth et al. 2007;
19. Pavlovski et al. 2014; 20. Popper 1984; 21. Popper et al. 1985; 22.
Andersen & Gimenez 1985; 23. Lacy & Popper 1984; 24. Andersen et al. 1984; 25.
Williamon et al. 2004

##  3 Data Reduction

Spectra were retrieved from the Hubble Space Telescope (HST) Mikulski Archive
for Space Telescope (MAST) archive, having been processed with the NEWSIPS
pipeline (Nichols and Linsky 1996). Further processing was done with the IUE
Regional Analysis Facility (IUERDAF) software package installed at the Harvard
& Smithsonian Center for Astrophysics, High Energy Division. The correction to
the IUE fluxes derived by Bohlin and Bianchi (2017) to match the HST CALSPEC
files (e.g. Space Telescope Imaging Spectrograph [STIS] spectra) was applied
to the IUE fluxes.

IUE NEWSIPS pipeline already removes known features such as the reseaux marks
on the camera and cosmic ray hits. In addition, for most exposure times
geocoronal Lyman $\alpha$ emission is present in the line cores, which was
removed (“blemished”) by eye. The spectra were then dereddened using the
reddening law of Cardelli, Clayton, and Mathis (1989) using the IDL procedure
unred_ccm.pro. E(B-V) was taken from TAG (except for WX Cep) and is listed in
the summary table (Table 6). The reddening for WX Cep was taken from Graczyk,
et al. (2019).

###  3.1 BOSZ Model Atmospheres

A grid of stellar atmospheres has been computed by Bohlin et al. (2017) from
the Kurucz Atlas9 code, which are available in the MAST archive.

To cover a range of hot stars, we used an initial 58 models spanning a
temperature range of 5,000–35,000 K. For the BOSZ models used in this work, we
chose the following initial conditions appropriate to main sequence stars in
IUE low resolution spectra: surface gravity (log g) = 4.0, solar metallicity
[M/H] = 0, solar carbon abundance [C/M] = solar, alpha abundance [alpha/M] =
0, instrumental broadening = 500 km/s, and microturbulence = 2 km/s. The
selection of these parameters was to reflect the characteristics of the DEBs
and IUE spectra as closely as possible, while keeping the analysis uniform for
all systems.

While the BOSZ temperature models span a large temperature range, the
increments at which a temperature model is available varies throughout the
range. For cooler temperatures, models are available every 250 K. For warmer
temperatures, models are available every 500 K. At the hottest temperatures,
models are available every 1000 K. To determine more precise temperature
uncertainties, we generated linearly interpolated models from the original
BOSZ models.

##  4 Analysis

The goal of this study was to determine a temperature which best reproduced
the spectrum from 1150 to 1975 Å within the framework of the models. The most
important aspects of the analysis approach are:

$\bullet$ The composite spectrum of the binary was modeled with a combination
of spectra from the models of the hot and cool components.

$\bullet$ The hotter star in the binary dominates at the shortest wavelengths,
and only that temperature is determined from the fits.

$\bullet$ The analysis is based on best-determined parameters from the eclipse
solution: the mass M, the radius R, and the ratio of temperatures of the two
stars T1/T2 where 1 and 2 refer to the primary (most massive) and secondary of
the binary system.

The details of the matching process are described in the sections below.

###  4.1 Spectrum–Model Comparisons

The comparison between the IUE spectrum and the BOSZ models was made as
follows. The radii of the primary and secondary stars in the systems are well
determined from the eclipse solutions. The analysis was started with the
temperatures T1 and T2 from spectral types from TAG. A model atmosphere was
selected close to the temperature for each star, and then the model of the
secondary was scaled using $(R_{2}/R_{1})^{2}$ from the eclipse solutions.
These two models were then summed and normalized to the flux of the IUE
spectrum. For the hottest stars, the Ly $\alpha$ region (1180 to 1250 Å) was
excluded because of contamination by interstellar Ly $\alpha$ absorption. The
summed model flux and the IUE flux were then used to calculate a standard
deviation (SD). The hottest star (generally the primary) dominates heavily at
the shortest wavelengths, so a series of comparisons were made stepping the
temperature of the primary T1 through a series of temperatures bracketing the
temperature expected from TAG. These comparisons were used in two ways. First,
the comparison was inspected as in Fig. 1. In general, this inspection showed
differences between models of T1 of 100 to 200 K for the cooler primaries and
500 K for hotter primaries. Second the difference between the composite model
and the IUE spectrum was formed and plotted through the sequence of T1 (Fig.
2). Again, in general, differences were clear between temperatures (T1) of
about 200 K, particularly at the long and short wavelength regions of the
spectra. The standard deviations produced a parabola as a function of T1 (Fig.
3), from which the minimum was used to determine T1.

Visual inspection of the spectral comparison and the difference plots was one
approach to temperature determination. In this case, an uncertainty was
estimated from the temperatures which could be identified by eye as too high
and too low to be matches ($\delta$ T1). The temperature was assumed to be
between these temperatures, or a quarter of $\delta$ T1.

We also tried a $\chi^{2}$ approach, using the IUE instrumental error.
However, because the study covered a large range of temperatures from 30,000
to 7,000 K two effects complicated the $\chi^{2}$ estimation. First, different
wavelength regions were very differently exposed on the camera, and hence had
very different distributions of actual uncertainties. Second the line opacity
is very different over that temperature range. Both these were difficult to
incorporate into the analysis. Standard deviations seemed to give more
consistent temperature uncertainties throughout the range.

Spectral comparisons such as Fig. 1 show the full spectrum. However, portions
of the spectrum which are saturated are marked in the upper left spectrum, as
are reseau marks, and geocoronal Ly $\alpha$. While these regions are shown in
the figure, they are omitted from the SD calculation. In addition in cool
spectra, there are features in the models near 1550Å, and 1700Å. While these
appear in the models which have complex opacity, they are not seen in any of
the spectra, so they are omitted from the fitting (Fig. 4).

###  4.2 Composite Spectra: Groups

The detailed analysis was done in four groups.

  * •

Group I: The primary and secondary have fairly similar temperatures

  * •

Group II: The secondary is markedly cooler than the primary

  * •

Group III: The secondary is actually hotter than the primary, i.e. the primary
has evolved off the main sequence

  * •

Group IV: What is it?

Group II: These are the easiest to analyze. Because the secondary is
significantly cooler than the primary, once the model flux is scaled
$(R_{2}/R_{1})^{2}$, the contribution of the secondary is very small. This
means that a modest change in temperature T2 will have an insignificant effect
on the summed model spectrum. Hence the procedure stepping through T1 values
will produce an optimal value of T1.

Group I: The analysis is a little more complicated. It begins as for Group II
to determine a T1 using an assumed T2. (Solutions for which T2 was larger than
T1 were discarded.) An additional constraint was then imposed. Since the
temperature ratio T1/T2 is well determined from the eclipse solution, it can
be used to obtain T2 from the value of T1 determined in the first step. With
this new value of T2 the process is repeated until a solution converges near
the T1/T2 from the eclipse solution. Occasionally, the temperatures of the two
components are determined to be indistinguishable.

Group III: This group similarly requires iteration, although in this case the
ratio of the radii brings the models of the two temperatures closer together,
complicating the determination of the temperature of the hotter star.

Group IV: This group contains puzzling or otherwise poorly defined solutions.

###  4.3 Examples

The analysis process is illustrated with examples from a hot star in the
sample V451 Oph, and a cool star YZ Cas, both in Group II.

V451 Oph The spectrum/model comparison, the difference comparison, and
standard deviation parabola are shown in Figs. 1, 2, and 3 respectively. In
Fig 2 since the temperatures for the primary 11500 and 12500 K are clearly
ruled out, an estimate of the “visual” uncertainty for T1 is 250 K.

Figure 1: V451 Oph: Comparison of the IUE unreddened spectrum and models.
Lines are: IUE spectrum: black; hot model: green; cool spectrum: blue; model
composite: red. Model T1 (hot) temperatures are indicated in each panel. The
best fit is for T1 = 12000 K. Wavelength is in Å; flux is in ergs cm-2 s-1 Å-1
in all figures.  Figure 2: V451 Oph: The difference between the composite
model and the spectrum. The differences from models of six T1 temperatures
(indicated in the figure) are shown from top to bottom. An arbitrary constant
has been added to the difference spectra so that they do not overlap. (T2
fixed at 9800 K.) The best fit is indicated with a black line. The changes as
a function of model temperature are clear in the comparison. The coolest model
(top) is too low in flux for the shorter wavelengths as compared with the
(normalized) longer wavelengths. Conversely, for a hotter model (bottom) the
model fluxes are larger than the spectrum for wavelengths shorter than 1300 Å.
In this temperature range, the poor agreement in the Ly $\alpha$ region near
1200 Å is particularly marked for the hottest model.  Figure 3: V451 Oph: The
temperature determination: the standard deviation as a function of temperature
of the primary. The temperature of the secondary was fixed at 11000 K; dots:
the standard deviation from the spectrum-model comparison; triangles: the
parabola fit.

YZ Cas As an example of a cooler star YZ Cas is used. The spectrum/model
comparison, the difference comparison, and the standard deviation parabola are
shown in Figs. 4, 5 and 6. From Fig. 5, the temperatures 8750 and 9250 K are
ruled out, making an estimate on the visual uncertainty of 125 K for T1. The
temperature determination from the SD parabola is shown in Fig. 6

Figure 4: YZ Cas: Comparison of IUE spectrum and models. Lines are the same as
in Fig 1. The magenta line above the spectrum in the upper left at 1550 Å
indicates that this feature was not included in the fit. Best fit is for T1 =
9000 K.  Figure 5: YZ Cas: The difference between the model and the IUE
spectrum. The differences from models of five temperatures is shown from top
to bottom with increasing model temperature. The best fit is indicated with a
black line.  Figure 6: YZ Cas: The temperature determination: the standard
deviation as a function of temperature of the primary. The temperature of the
secondary was fixed at 7250 K; dots: the standard deviation from the spectrum-
model fit; triangles: the parabola fit

###  4.4 Sources With One-Temperature Fits

For PV Cas, the temperatures, masses, and radii of both the primary and
secondary are assumed to be the same (TAG). In this case, only one temperature
T1 was fit to the model. A 1T fit was also explored in other cases,
specifically for CV Vel, QX Car, and VV Pyx.

###  4.5 Reddening Tests

One of the first concerns in working with stellar energy distributions (SEDs)
in the ultraviolet is the effect of reddening on derived temperatures. While
the ultraviolet fluxes are heavily affected by reddening or uncertainty in
reddening, temperatures derived from ultraviolet SEDs are surprisingly robust
to changes in reddening, at least for the temperatures of the stars that we
are working with. This is for two reasons. First the wavelength range we are
working with is only 750 Å (1150 to 1900 Å), making the variation of flux with
a change in E(B-V) within that wavelength range quite small. Second, the
observed energy distributions have a well defined shape in flux versus
wavelength so that the comparison with the models is fairly tightly defined.
For example, the hottest stars typically have flux which decreases sharply at
shorter wavelengths.

We have tested the sensitivity of our temperature fits to reddening on a range
of E(B-V) values for V451 Oph (Table 3). The uncertainties in Table 3 are from
the visual estimates as discussed in Sect. 4.1. Although the range in E(B-V)
is much larger than possible uncertainties in E(B-V), the resulting
temperatures only differ by $\pm$ 450 K or $\pm$ 4%. Thus for a typical
uncertainty in E(B-V) (perhaps 0.03 mag) the uncertainty in the temperature is
$<$1%. In addition, most values of E(B-V) for the targets are very small
(Table 6), implying small uncertainties in the reddening and temperature.

Table 3: E(B-V) Tests for V451 Oph  E(B-V) | T | $\pm$
---|---|---
(mag) | (K) | (K)
0.05 | 11734.9 | 125
0.10 | 11933.4 | 125
0.158 | 12161.8 | 250
0.20 | 12376.3 | 312
0.25 | 12639.4 | 280

###  4.6 M-K Standards

In order to explore the sensitivity of the fits to IUE spectra of early B
stars, three spectra from the IUE Ultraviolet Spectral Atlas (Wu, et al. 1983)
of Morgan-Keenan spectral type standards (or similar substitutes where
necessary) were analyzed in the same way as the DEBs. Results are provided in
Appendix A. For comparison temperatures for several calibrations of the MK
classes are provided.

###  4.7 Temperature Summary

The temperature results are summarized by Group in Table 4. The columns list
the star and T1 from the parabola fit. The two error estimates are from the
standard deviation parabola and the visual inspection of the spectrum/model
comparison. T2 is the temperature of the secondary in the fit, starting from
the value in TAG, but adjusted if required by the comparisons. The T2/T1 ratio
from TAG is in the next column. In borderline cases there is a little overlap
in this ratio between Group I and Group II. The final column provides notes to
our fits. For PV Cas a one-temperature fit was made (Sect. 4.4). For Group III
(bottom) T2 is the hotter temperature and is listed in Column 2 rather than
Column 5.

One of the goals of this study was to determine the temperature range for
which IUE spectra provide accurate temperatures. This is discussed further in
Section 6.

Table 4: Temperatures by Group

Star | T1 | Err | Err | T2 | T2/T1 a | Notes b
---|---|---|---|---|---|---
|  | SD | Vis |  |  |
| (K) | (K) | (K) | (K) |  |
Group II |  |  |  |  |  |
V539 Ara | 19248 | 363 | 500 | 17500 | 0.945 |
GG Lup | 15793 | 1518 | 250 | 11500 | 0.746 |
$\zeta$ Phe | 14483 | 2440 | 250 | 12000 | 0.833 |
IQ Per | 13241 | 3051 | 250 | 8750 | 0.626 |
TZ Men | 10418 | 1123 | 100 | 7250 | 0.692 |
V451 Oph | 11980 | 130 | 250 | 11000 | 0.907 |
YZ Cas | 8982 | 385 | 125 | 7250 | 0.706 |
V624 Her | 8160 | 335 | 125 | 8000 | 0.975 |
GZ CMa | 8566 | 440 | 125 | 8250 | 0.966 |
EE Peg | 8687 | 119 | 125 | 6500 | 0.741 |
Group I |  |  |  |  |  |
QX Car | 23428 | — | 1000 | 23000 | 0.950 |
CV Vel | 18045 | 563 | 500 | 18000 | 0.989 |
V760 Sco | 17092 | 854 | 500 | 16000 | 0.964 |
PV Cas | 10632 | 677 | 250 |  | 0.999 | 1T
$\beta$ Aur | 9177 | 914 | 125 | 9000 | 0.984 |
V1647 Sgr | 9361 | 535 | 125 | 8750 | 0.948 |
VV Pyx | 9751 | 782 | 125 | 9500 | 1.00 |
Group IV |  |  |  |  |  |
$\chi^{2}$ Hya | 12160 | 1370 | 250 | 11500 | 0.945 |
Group III |  |  |  |  |  |
| T2 | Err | Err | T1 |  |
|  | SD | Vis |  |  |
WX Cep | 9087 | 1092 | 125 | 8375 | 1.08 |
V1031 Ori | 8906 | 419 | 100 | 7750 | 1.07 |
AY Cam | 7590 | 318 | 150 | 7200 | 1.02 |

Note. — a The T2/T1 is from TAG; b 1T: one temperature model fit

##  5 Comments on Individual Systems

###  5.1 V478 Cyg

The temperature for V478 Cyg has one of the largest uncertainties in this
study for two reasons. First, it is one of the weakest exposures. In addition
to that the E(B-V) is somewhat uncertain. The E(B-V) in TAG is 0.85, taken
from Popper and Etzel (1981). As they discuss, the B-V measurement of the
system (Popper and Dumont 1977) was done with two telescopes (Palomar and Kitt
Peak). The larger diaphragm at Palomar included an additional star. This star
is 4.26 mag fainter than V478 Cyg, hence would have only a very small effect
on the color, most likely making it slightly redder. Since photometry of the
system is typically made differentially, measurements of B-V are scarce. We
were not able to achieve a definitive fit, and omit V478 Cyg from further
consideration.

###  5.2 AH Cep

The IUE spectrum of AH Cep is saturated for half the wavelength region. For
this reason, the standard temperature fitting is unreliable. AH Cep is omitted
from Mass-Temperature fitting.

###  5.3 U Oph

Although the exposure of U Oph is reasonably good, the spectrum is saturated
in several important wavelength regions. For this reason, it is omitted from
temperature discussions.

###  5.4 QX Car

The temperatures of the two components are similar (Group I), but the masses
are clearly not identical (TAG). We have investigated both 1T and 2T fits. The
temperature from the 2T fit is 23430 K, with an estimated visual uncertainty
of 1000 K.

###  5.5 V578 Mon

Because the spectrum rises monotonically toward short wavelengths, the
temperature and reddening are degenerate, therefore the temperature from the
IUE spectrum is not definitive, and is not used in subsequent temperature
discussions. Subsequent to TAG, a small adjustment was made to the ratio of
radii (Garcia, et al. 2013), resulting in a difference of the normalization
factor (R2/ R1)2 of 2%, which has no effect on T1.

###  5.6 V539 Ara

The IUE observation occurred just inside the start of primary eclipse.
However, V539 Ara is a Group II system, with the secondary significantly less
luminous than the primary in the ultraviolet. The decrease in flux in the
primary of approximately 5% will have minimal effect on the spectral fitting.
(Fig. 1 illustrates a similar system.)

###  5.7 Group III: WX Cep, and V1031 Ori, and AY Cam

For the three stars in this group the hotter star is actually the secondary
which is less luminous, and smaller, so the more massive star is less dominant
in the ultraviolet. For this reason it is more difficult to disentangle the
composite spectrum, and the temperature of the hotter star is less well
determined. We include estimates of the temperatures in Tables 4 and 6 but
they are omitted from the Mass-Temperature relation determination.

###  5.8 Group IV: $\chi$2 Hya

In this system, the primary is significantly evolved beyond the main sequence
(Clausen and Nordstrom 1978). For this reason, it is not used in the main
sequence Mass-Temperature relation determination.

##  6 Hot Stars

One of the aims of this project was to explore the upper range of temperature
where IUE spectra provide definitive temperatures. The hottest stars in the
sample (V478 Cyg, AH Cep, and V578 Mon) did not produce well determined
temperature (Section 5). U Oph also was unsatisfactory because of the
overexposed spectrum

Ribas et al. (2000) have determined the temperatures from photometry, compared
with Kurucz atmospheres. In order to check the upper temperature limit for
which the IUE spectra are accurate, the temperature determinations of the two
hottest stars in the IUE DEB sample are compared with the Ribas et al.
temperatures in Table 5. The temperatures of the two hottest stars agree
within the errors.

Table 5: Temperatures of Early B Stars  | Spect | T | T
---|---|---|---
| Type | Table 4 | Ribas et al.
|  |  | (2000)
|  | (K) | (K)
QX Car | B2 V | 23428 $\pm$ 1000 | 24831 $\pm$ 520
CV Vel | B2.5 V | 18045 $\pm$ 500 | 17947 $\pm$ 500

##  7 Discussion

###  7.1 Binary Parameter Summary

Parameters for the DEB systems are summarized in Table 6. Each system has two
rows, one for the primary and one for the secondary. The first entry is the
name of the system (line 1) and the HD or BD number (line 2). The next columns
are the spectra type, the mass and its error in $M_{\odot}$, the temperature
and its error in K, the radius and its error in $R_{\odot}$, the E(B-V) and
its error (where known) in mag, the temperature and error from this study
(Table 4), and the log luminosity in solar units (Section 7.7). The mass,
spectral type, radius, temperature, and E(B-V) are from TAG. E(B-V) for WX Cep
is from Graczyk et al (2019).

Table 6: Parameters for the DEB Sample  System | SpecType | Mass | $\pm$ | T | $\pm$ | Radius | $\pm$ | E(B-V) | $\pm$ | T | $\pm$ | Log Lum
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | TAG | TAG |  |  |  |  | IUE | IUE |
|  | ($M_{\odot}$) | ($M_{\odot}$) | (K) | (K) | ($R_{\odot}$) | ($R_{\odot}$) | (mag) | (mag) | (K) | (K) | ($L_{\odot}$)
V478 Cyg | O9.5V | 16.62 | 0.33 | 30479 | 1000 | 7.426 | 0.072 | 0.85 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 193611 | O9.5V | 16.27 | 0.33 | 30549 | 1000 | 7.426 | 0.072 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
AH Cep | B0.5Vn | 15.26 | 0.35 | 29900 | 1000 | 6.346 | 0.071 | 0.58 | 0.08 |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 216014 | B0.5Vn | 13.44 | 0.25 | 28600 | 1000 | 5.836 | 0.085 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V578 Mon | B1V | 14.50 | 0.12 | 30000 | 740 | 5.149 | 0.091 | 0.455 | 0.029 |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 259135 | B2V | 10.262 | 0.084 | 26400 | 600 | 4.21 | 0.10 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
QX Car | B2V | 9.25 | 0.12 | 23800 | 500 | 4.291 | 0.091 | 0.049 |  $\cdots$ | 23428 | 1000 | 3.6701
HD 86118 | B2V | 8.46 | 0.12 | 22600 | 500 | 4.053 | 0.091 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V539 Ara | B3V | 6.24 | 0.066 | 18100 | 500 | 4.516 | 0.084 | 0.071 |  $\cdots$ | 19248 | 500 | 3.3828
HD 161783 | B4V | 5.314 | 0.06 | 17100 | 500 | 3.428 | 0.083 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
CV Vel | B2.5V | 6.086 | 0.044 | 18100 | 500 | 4.089 | 0.036 | 0.035 |  $\cdots$ | 18045 | 500 | 3.1535
HD 77464 | B2.5V | 5.982 | 0.035 | 17900 | 500 | 3.950 | 0.036 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
U Oph | B5V | 5.273 | 0.091 | 16440 | 250 | 3.484 | 0.021 | 0.226 | 0.007 |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 156247 | B6V | 4.739 | 0.072 | 15590 | 250 | 3.110 | 0.034 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V760 Sco | B4V | 4.969 | 0.09 | 16900 | 500 | 3.015 | 0.066 | 0.33 |  $\cdots$ | 17092 | 500 | 2.8112
HD 147683 | B4V | 4.609 | 0.073 | 16300 | 500 | 2.641 | 0.066 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  |
GG Lup | B7V | 4.106 | 0.044 | 14750 | 450 | 2.380 | 0.025 | 0.027 |  $\cdots$ | 15793 | 250 | 2.3995
HD 135876 | B9V | 2.504 | 0.023 | 11000 | 600 | 1.726 | 0.019 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$\zeta$ Phe | B6V | 3.921 | 0.045 | 14400 | 800 | 2.852 | 0.015 | 0 |  $\cdots$ | 14483 | 250 | 2.5539
HD 6882 | B8V | 2.545 | 0.026 | 12000 | 600 | 1.854 | 0.011 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$\chi^{2}$ Hya | B8V | 3.605 | 0.078 | 11750 | 190 | 4.390 | 0.039 | 0.016 |  $\cdots$ | 12160 | 250 | 2.4619
HD 96314 | B8V | 2.632 | 0.049 | 11100 | 230 | 2.159 | 0.030 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
IQ Per | B8V | 3.504 | 0.054 | 12300 | 230 | 2.445 | 0.024 | 0.14 | 0.01 | 13241 | 250 | 2.1479
HD 24909 | A6V | 1.730 | 0.025 | 7700 | 140 | 1.499 | 0.016 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  |
PV Cas | B9.5V | 2.816 | 0.05 | 10200 | 250 | 2.301 | 0.020 | 0.217 |  $\cdots$ | 10632 | 250 | 1.7687
HD 240208 | B9.5V | 2.757 | 0.054 | 10190 | 250 | 2.257 | 0.019 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V451 Oph | B9V | 2.769 | 0.062 | 10800 | 800 | 2.642 | 0.031 | 0.158 |  $\cdots$ | 11980 | 250 | 1.9555
HD 170470 | A0V | 2.351 | 0.052 | 9800 | 500 | 2.029 | 0.028 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
WX Cep | A5V | 2.533 | 0.05 | 8150 | 250 | 3.996 | 0.03 | 0.19 | 0.03 |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 213631 | A2V | 2.324 | 0.045 | 8900 | 250 | 2.712 | 0.023 |  $\cdots$ |  $\cdots$ | 9087 | 125 |  $\cdots$
TZ Men | A0V | 2.482 | 0.025 | 10400 | 500 | 2.017 | 0.02 | 0 |  $\cdots$ | 10418 | 100 | 1.6741
HD 39780 | A8V | 1.500 | 0.010 | 7200 | 300 | 1.433 | 0.014 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V1031 Ori | A6V | 2.468 | 0.018 | 7850 | 500 | 4.323 | 0.034 | 0.034 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 38735 | A3V | 2.281 | 0.016 | 8400 | 500 | 2.978 | 0.064 |  $\cdots$ |  $\cdots$ | 8906 | 100 |  $\cdots$
$\beta$ Aur | A1m | 2.375 | 0.027 | 9350 | 200 | 2.765 | 0.018 | 0 |  $\cdots$ | 9177 | 125 |  $\cdots$
HD 40183 | A1m | 2.304 | 0.030 | 9200 | 200 | 2.571 | 0.018 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
YZ Cas | A1m | 2.317 | 0.020 | 10200 | 300 | 2.539 | 0.026 | 0.07 |  $\cdots$ | 8982 | 125 | 1.7423
HD 4161 | F2V | 1.352 | 0.009 | 7200 | 300 | 1.350 | 0.014 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V624 Her | A3m | 2.277 | 0.014 | 8150 | 150 | 3.031 | 0.051 | 0.05 | 0.01 | 8160 | 125 | 1.5924
HD 161321 | A7m: | 1.876 | 0.013 | 7950 | 150 | 2.210 | 0.034 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
GZ CMa | A3m | 2.199 | 0.017 | 8800 | 350 | 2.494 | 0.031 | 0.07 | 0.03 | 8566 | 125 | 1.4951
HD 56429 | A4V: | 2.006 | 0.012 | 8500 | 350 | 2.132 | 0.037 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
V1647 Sgr | A1V | 2.184 | 0.037 | 9600 | 300 | 1.832 | 0.018 | 0.04 |  $\cdots$ | 9361 | 125 | 1.4346
HD 163708 | A1V | 1.967 | 0.033 | 9100 | 300 | 1.667 | 0.017 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
EE Peg | A3m | 2.151 | 0.024 | 8700 | 200 | 2.090 | 0.025 | 0 |  $\cdots$ | 8687 | 125 | 1.2871
HD 206155 | F5V | 1.332 | 0.011 | 6450 | 300 | 1.312 | 0.013 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
VV Pyx | A1V | 2.097 | 0.022 | 9500 | 200 | 2.168 | 0.02 | 0.022 |  $\cdots$ | 9751 | 125 | 1.7809
HD 71581 | A1V | 2.095 | 0.019 | 9500 | 200 | 2.168 | 0.02 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
AY Cam | A0V | 1.905 | 0.04 | 7250 | 100 | 2.772 | 0.02 | 0.06 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
BD+77 328 | F0V | 1.707 | 0.036 | 7395 | 100 | 2.026 | 0.017 |  $\cdots$ |  $\cdots$ | 7590 | 150 |  $\cdots$

###  7.2 Comparison between TIUE and TTAG

The comparison between the log of the IUE temperatures derived here from
spectral fits and the log of the temperatures from TAG is shown in Fig. 7. In
general they are similar within the errors, with an insignificant offset. The
line shows the fit from the least squares bisector (unweighted), as
recommended by Isobe, et al. (1990):

log T(TAG) = 0.104 $\pm$ 0.114 $+$ (0.973 $+$ 0.020) $\times$ log T(IUE) (Eqn
1)

Figure 7: The temperatures from the IUE spectral fit Log T(IUE) versus the
temperatures from TAG Log T(TAG). Both temperatures are in K.

###  7.3 The M-R Relation: Evolution

The Log Mass–Log Radius relation from TAG data is shown in Fig. 8. Stars with
radii larger than the main sequence minimum indicates evolution beyond the
Zero Age Main Sequence, which is a major source of scatter in the Mass-
Temperature relation. The stars marked in red in Fig. 8 are from several
groups identified in this study. They include $\chi^{2}$ Hya, which is known
to be evolved. Group III stars are also evolved. In this case it is the
temperature of the secondary which we determine. The primary is clearly
evolved in Group III systems, but the secondary may also be. In fact, for all
three stars (WX Cep, V1031 Ori and AY Cam), the radius is larger than the
radii for the main sequence. The 5 stars with an Am (metallic) primary
($\beta$ Aur, YZ Cas, V624 Her, GZ CMa, and EE Peg) are discussed in the next
section. All these stars shown in red sit above the main sequence relation in
Fig. 8, and account for most of the spread in the Log M–Log R relation. This
is discussed further in the next section.

Figure 8: Log Radius vs Log Mass relation from the TAG sample. Main sequence
stars are in black; other stars are in red. The group of other stars includes
$\chi^{2}$ Hya, Group III stars, and Am stars (see text). Masses are in
$M_{\odot}$; radii are in $R_{\odot}$

###  7.4 Metallicity

Five stars in the DEB sample have a metallic Am primary ($\beta$ Aur, YZ Cas,
V624 Her, GZ CMa, and EE Peg). In the previous section and Fig. 8, it was
shown that they have larger radii (for a given mass) than normal A stars. In
this case, they are not necessarily evolved, simply showing the effects of a
metal-rich atmosphere. In the mass-temperature relation in the next section,
they typically have cooler temperatures than normal A stars coupled with their
larger radii. Again, this is explained by a metal-rich atmosphere. In
addition, the BOSZ models used here assume solar abundances, and hence are not
appropriate for Am stars. For this reason these stars are omitted from the
determination of the Mass-Temperature relation.

###  7.5 The M-T Relation

The relation between mass and temperature is shown in Fig. 9. (The errors in
the temperatures are from the visual inspection.) The systems plotted in black
are used to determine the relation (omitting stars showing evolution and
chemically peculiar stars plotted in red).

Figure 9: The Mass -Temperature relation. Main sequence stars: black; others:
red. Others include $\chi^{2}$ Hya, Group III stars, and Am stars. Masses are
in $M_{\odot}$; temperatures are in K.

For this limited temperature (23000 to 8000K) and mass range, we have assumed
a linear relation between log M and log T. The relation from bisector least
squares is:

log M/$M_{\odot}$ = -5.90 $\pm$ 0.27 + (1.56 $\pm$ 0.07) x log T (Eqn 2)

The RMS of the residuals from the fit of the log M/$M_{\odot}$ is 0.037.

This mass–temperature relation is directly determined. However, there are
several factors which contribute to the width of the relation. Different
rotation velocities and different abundances will result in a spread in
temperatures between stars. In addition during their main sequence lifetimes,
stars cool, resulting in a range of temperatures. As an example, the
evolutionary tracks from Ekström, et al. (2012) span approximately 0.1 dex in
Log T during the main sequence. This corresponds to an uncertainty of
approximately 0.08 dex in log M or about 18%. The scatter about the mean
relation ($\pm$ 0.04 dex in log M) due to evolution within the main sequence
is about the same as the RMS around equation 2. That is, the age spread
accounts for at least a large part of the RMS scatter.

In certain cases, this age-related uncertainty is lower. For instance, a
secondary which is the companion of a more massive star must be comparatively
young. The hot companion of a Cepheid is an example which must be close to the
zero age main sequence (ZAMS). That is, it will be on the hotter edge of the
M-T relation with a smaller uncertainty than a random main sequence star.

###  7.6 Comparisons Against Other M-T Relations

The linear fit to the main sequence points (black points in Fig. 9) is shown
in Fig. 10 in the dashed line. Harmanec (1988) used the sample of DEBs
available at that time to derive a Mass-Temperature relation over the full
length of the main sequence. The solid line shows the relation from Harmanec
(1988). The Harmanec relation is curved since it describes a much larger
temperature range. However the agreement is essentially good in this
temperature range.

Figure 10: The Mass -temperature relation. Main sequence stars: black points;
dashed line: fit to points; solid line: Harmanec relation.

###  7.7 The Luminosity–Temperature Relation

For completeness we provide the Log Temperature –Log Luminosity relation in
Fig. 11. Luminosities are derived from TAG values scaled by
$(T_{IUE}/T_{TAG})^{4}$. The stars shown omit systems showing evolution
($\chi^{2}$ Hya, WX Cep, V1031, and AY Cam), as well as Am stars.

Figure 11: The Log Temperature–log Luminosity relation for main sequence stars
showing no evolution or chemical peculiarities (see text). Luminosity is in
solar units; the temperature is determined from the IUE spectral fits.

##  8 Conclusions

This study combines masses for DEBs with temperatures determined from
ultraviolet IUE spectra and BOSZ model atmospheres. For this important group
on which the calibration of masses on the main sequence rests, a direct Mass-
Temperature relation is determined for B and early A stars. The mass-radius
plot (Fig 8) shows that some of the sample is evolved beyond the main
sequence. These stars as well as chemically peculiar Am stars have been
removed from the calibration sample. The Mass-Temperature relation is probably
still somewhat broadened by variation in evolution and abundance, however Fig.
9 demonstrates that the scatter is relatively small (RMS of the log
M/$M_{\odot}$ is 0.037). The comparison between the IUE temperatures and those
in TAG from spectral types is good. Furthermore, the Mass-Temperature relation
is very similar to that derived by Harmanec (1988).

##  9 Acknowledgments

The authors would like to thank Dr. Hans Moritz Günther for his insightful
comments and assistance regarding our temperature fitting. We thank the
referee for comments that improved the content of the paper as well as the
presentation. It is a pleasure to thank Dr. Richard Gray for comments on
chemically peculiar stars. M.G.F would like to thank Dr. Matt Ashby and Dr.
Jonathan McDowell for their invaluable comments and guidance during the SAO
REU program. The SAO REU program is funded in part by the National Science
Foundation REU and Department of Defense ASSURE programs under NSF Grant no.
AST-2050813, and by the Smithsonian Institution. Support was provided to
N.R.E. by HST Grant GO-15861.001-A and by the Chandra X-ray Center NASA
Contract NAS8-03060. This research is based on observations made with the IUE
satellite, obtained from the MAST data archive at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5–26555. This research has made
use of the SIMBAD database, operated at CDS, Strasbourg, France.

##  10 Appendix A: Early B Stars

In order to confirm the temperature sensitivity of the IUE spectra of the
hotter stars in the DEB sample we examined stars from the IUE Spectral Atlas
(Wu, et al. 1983). The IUE Atlas project observed samples of the full range of
the Morgan-Keenan (MK) spectral system in low dispersion in both the short and
long wavelength regions, primarily using stars defining the MK classes. Table
7 lists the spectra selected for this test for spectral classes B1 V, B3 V,
and B5 V. Columns provide the spectral type, the HD number, V, E(B-V), the IUE
spectrum number, the exposure level, alternate designations, MK for stars
which define MK classes, and the temperature, and the temperature uncertainty
from the SD comparisons, and from visual comparisons. The maximum exposure is
in instrumental Data Numbers (DN), where an optimal exposure is near 200 DN.
The spectra were fitted to BOSZ models (1T fit). The resulting temperatures
are listed in Table 7. The final three columns list temperatures for these
classes for 3 widely used calibrations of temperatures for MK classes:
Drilling and Landolt in Astrophysical Quantities (2000), Harmanec (1988) and
Pecaut and Mamajek (2013). The agreement is reasonable, given that MK classes
have some width in temperature. Lyubimkov et al. (2002) derive T=22500 $\pm$
600 K for HD31726, which is close to the IUE result

Table 7: IUE Observations of MK Standard Stars

Spectral | HD | V | E(B-V) | Spectrum | Exp Max | ID |  | T | $\pm$ | $\pm$ | AQ | Harm | PM
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Type |  |  |  |  |  |  |  |  | SD | Vis |  |  |
|  | (mag) | (mag) |  | (DN) |  |  | (K) | (K) | (K) | (K) | (K) | (K)
B1 V | 31726 | 6.15 | 0.05 | SWP8165 | 230 |  |  | 22471.0 | 908 | 500 | 24540 | 26180 | 26000
B3 V | 190993 | 5.06 | 0.02 | SWP9961 | 200 | 17 Vul | MK | 18025.1 | 401 | 500 | 19000 | 19050 | 17000
B5 V | 34759 | 5.22 | 0.02 | SWP15537 | 190 |  $\rho$ Aur | MK | 16373.3 | 378 | 375 | 15200 | 15490 | 15700

AQ: Drilling, and Landolt (2000)

Harm: Harmanec (1988)

PM: Pecaut and Mamajek (2013)

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# A Comparison of SOR, ADI and Multigrid Methods for Solving Partial
Differential Equations

Mohamed Mohsen Ahmed  Kansas State University, Manhattan, Kansas, USA

###### Abstract

This article presents several numerical techniques for solving Laplace
equation. A numerical FORTRAN solver is developed to solve the 2D laplace
equation. The numerical approaches implemented in the solver include Jacobi,
Gauss-Siedel, Successive Over Relaxation, Alternating Direct Implicit and
Multigrid methods. Detailed comparison between different numerical methods is
presented and discussed.

##  1 Nomenclature

$k$ = | iteration number
---|---
$L_{x}$ = | domain length in x direction
$L_{y}$ = | domain length in y direction
$m$ = | number of points in x direction
$n$ = | number of points in y direction
$R$ = | Residual
$w$ = | relaxation factor
$\psi$ = | stream function

##  2 Introduction

Laplace equation is an elliptic partial differential equation that governs
several simple physical problems such as irrrotational incompressible fluid
flow and steady state heat transfer in solids. The mathematical formula of a
two-dimensional Laplace equation is

$\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial
y^{2}}=0$ (1)

The finite difference representation of Laplace equation can be expressed in
two methods, the five point stencil and the nine point stencil. The geometric
interpretation of the two methods is provided in figure 1. The five point
stencil method is the most common, and its formula is obtained by

$\frac{\psi_{i-1,j}-2\psi_{i,j}+\psi_{i+1,j}}{\Delta
x^{2}}+\frac{\psi_{i,j-1}-2\psi_{i,j}+\psi_{i,j+1}}{\Delta x^{2}}=0$ (2)

This finite difference formula has a second order truncation error. On the
other hand, the nine point formula is obtained by

$\psi_{i+1,j+1}+\psi_{i-1,j+1}+\psi_{i+1,j-1}+\psi_{i-1,j-1}-2\frac{h^{2}-5k^{2}}{h^{2}+k^{2}}(\psi_{i+1,j}+\psi_{i-1,j})+2\frac{5h^{2}-k^{2}}{h^{2}+k^{2}}(\psi_{i,j+1}+\psi_{i,j-1})-20\psi_{i,j}=0$
(3)

This formula is also second order accurate in both spacial directions.
However, if $k=h$, the formula becomes of order six in both spacial
directions. We are concerned only with the five point stencil formula in the
current study.

The solution of the finite difference form of the Laplace equation is similar
to the solution of linear algebraic equations. There exist two methods in
order to solve any linear group of algebraic equation. The first method is
concerned with direct solution of the system of equation. These methods
include Crammer’s rule, Gauss elimination and Thomas algorithm of tri-diagonal
matrices. Although these methods are very simple, they require huge
computational time.

Iterative methods, on the other hand, are more efficient than direct methods
in most cases since they require far less computational time. In this study,
we present an extensive comparison between different iterative methods that
are commonly used to solve two-dimensional Laplace equation on a given
computational domain [1].

Figure 1: Five and nine point stencils

##  3 Problem Definition

The problem is concerned with two-dimensional steady incompressible irrational
flow in the chamber described in figure 2. The parameter $\psi$ here
represents the stream function of the potential flow. The inlet and outlet
small gaps are $CA=BD=0.25m$. A uniform grid is constructed with constant
spacing $\Delta x=\Delta y=0.25m$. The computational grid and boundary
condition is provided in figure 3. The stream function $\psi$ has a value of
zero on wall AB and zero on all other walls. The mesh size is obtained by

$m=\frac{L_{x}}{\Delta x}+1=25\qquad,\qquad n=\frac{L_{y}}{\Delta y}+1=17$ (4)

where $L_{x}$ and $L_{y}$ are the dimensions of the domain in $x$ and $y$
directions respectively. The total number of points in the domain is 425
points including 341 internal points and 84 boundary points.

Figure 2: Geometry description
Figure 3: Computational grid and boundary conditions

##  4 Iterative Methods

Iterative methods are used to solve linear algebraic equation by assuming an
initial value of the computed parameters and then applying the same algorithm
for certain number of calculations in order to converge to the final value of
the computed parameters. There exist two types of iterative methods, explicit
and implicit.

### Jacobi iterative method

Jacobi method is an explicit iterative method in which at any iteration cycle
every point in the domain is computed from the values of its neighboring
points at the old iteration cycle. The finite difference formula employed to
compute each point is

$\psi_{i,j}^{k+1}=\frac{\psi_{i+1,j}^{k}+\psi_{i-1,j}^{k}+\beta^{2}(\psi_{i,j+1}^{k}+\psi_{i,j-1}^{k})}{2(1+\beta^{2})}$
(5)

where $\beta=\frac{\Delta x}{\Delta y}$. Since each element of the right hand
side of this equation is at the same iteration cycle (i.e. old iteration), the
computations in this method can be parallelized. Therefore, Jacobi method is
said to be vectorizable.

### Gauss-Seidel method (GS)

At a current iterative cycle $k+1$ in the calculation of the point $i,j$ using
GS method, the neighboring points $i-1$ and $j-1$ that proceed the current
point $i,j$ are not considered at the old iteration cycle $k$. Their updated
values at the current iteration $k+1$ are used instead of their values at the
old iteration $k$.The finite difference formula that is used in GS iterations
is expressed as

$\psi_{i,j}^{k+1}=\frac{\psi_{i+1,j}^{k}+\psi_{i-1,j}^{k+1}+\beta^{2}(\psi_{i,j+1}^{k}+\psi_{i,j-1}^{k+1})}{2(1+\beta^{2})}$
(6)

GS iterative method is considered more efficient that Jacobi iterative method
in terms of the number of iterations. However, it might not be more efficient
in terms of computational time since it is not vectorizable.

### Successive over relaxation method (SOR)

In this method, a relaxation factor $w$ is used in order to accelerate the
iterative procedure. SOR method applied to Gauss-Seidel method is

$\psi_{i,j}^{k+1}=(1-w)\
u_{i,j}^{k}+w\frac{\psi_{i+1,j}^{k}+\psi_{i-1,j}^{k+1}+\beta^{2}(\psi_{i,j+1}^{k}+\psi_{i,j-1}^{k+1})}{2(1+\beta^{2})}$
(7)

I $1<w<2$ over relaxation is employed and the solution is accelerated. If
$w<1$ under-relaxation is employed which makes the solution more stable but
slower. If $w>2$ the solution might be unstable. The effect of $w$ on the
solution stability and speed is discussed in details in the results and
discussion section.

### Successive over relaxation by line (SLOR)

In this method, either rows or column are grouped together as shown in figure
4. The solution of each grouped row or column can be obtained implicitly using
Thomas Algorithm of tri-diagonal matrices. Figure 5 shows the flowchart of a
single SLOR cycle.

Figure 4: SLOR method Figure 5: Flowchart of single SLOR cycle

There are two ways in which relaxation can be employed to update the solution
to the next iteration, either before solving by Thomas algorithm (SLORB) or
after obtaining the solution from Thomas algorithm (SLORA). The finite
difference formula of the SLORA that is solved for each row at $k^{\prime}+1$
cycle using Thomas algorithm is

$-\psi_{i-1,j}^{k^{\prime}+1}+2(1+\beta^{2})\psi_{i,j}^{k^{\prime}+1}-\psi_{i+1,j}^{k^{\prime}+1}=\beta^{2}(\psi_{i,j+1}^{k}+\psi_{i,j-1}^{k+1}$
(8)

The solution over relaxation is obtained by

$\psi_{i,j}^{k+1}=\psi_{i,j}^{k}+w(\psi_{i,j}^{k^{\prime}+1}-\psi_{i,j}^{k})$
(9)

The finite difference equation for SLORB that is used to solve for each row at
$k+1$ cycle using Thomas algorithm is

$-2(1+\beta^{2})\psi_{i-1,j}^{k+1}+w\psi_{i,j}^{k+1}-2(1+\beta^{2})\psi_{i+1,j}^{k+1}=2(1+\beta^{2})(1-w)\psi_{i,j}^{k}+w\beta^{2}(\psi_{i,j+1}^{k}+\psi_{i,j-1}^{k+1})$
(10)

### Alternative direct implicit method (ADI)

ADI method is similar to SLOR method, but in each cycle sweeps by rows are
followed by sweeps by columns. Therefore, a complete iteration cycle consists
of two steps, as shown in the figure 6, which may require more computational
time than SLOR method.

Figure 6: Flowchart of ADI method

### Multigrid Method (V-Cycle)

On fine grids, very large number of iteration may be required to solve Laplace
equation using the previously discussed methods. This large number of
iteration is mostly cosed by the low frequency component of the error, which
can not easily be removed on fine grids. therefore, coarse grids are used in
mutligrid method in order to remove the low frequency error. Let the linear
operator of the Laplace equation be

$L(\psi_{i,j})=\frac{\psi_{i-1,j}-2\psi_{i,j}+\psi_{i+1,j}}{\Delta
x^{2}}+\frac{\psi_{i,j-1}-2\psi_{i,j}+\psi_{i,j+1}}{\Delta x^{2}}$ (11)

If a solution is obtained using GS method after only three of four iteration
(i.e. non converged solution), the right hand side of this equation does not
equal zero. And the residual can be obtained by

$R_{i,j}=L(\psi_{i,j})$ (12)

such that $R_{i,j}=0$ at convergence. Let the final converged solution be
$\psi_{i,j}$ defined by the correction

$\psi_{i,j}=\Delta\psi_{i,j}+\psi_{i,j}^{k}$ (13)

where $\Delta\psi_{i,j}$ is the difference between the last two iterations and
$\psi_{i,j}^{k}$ is the solution obtained from iteration $k$. Since
$L\psi_{i,j}=0$, we can write

$L\Delta\psi_{i,j}+L\psi_{i,j}^{k}=0$ (14)

Using equation 12 we obtain

$L\Delta\psi_{i,j}=-R_{i,j}$ (15)

Three mesh levels are considered in the multigrid V-cycle. Mesh level 1 has
25x17 points, mesh level 2 has 13x9 points and mesh level 3 has 7x5 points.
The procedure of the V-cycle is shown in 7. The solution procedure employed in
this study is as follows:

  1. 1.

Mesh level 1: solve $L(\psi^{1}_{i,j})$=0 for 3 iterations by GS method using
initial condition $(\psi_{i,j}=0)$

  2. 2.

Mesh level 1: compute $R^{1}_{i,j}=L(\psi_{i,j}^{1})$

  3. 3.

Mesh level 2: restrict the values of $R^{1}_{i,j}$ from mesh level 1 to the
coarse grid in mesh level 2

  4. 4.

Mesh level 2: solve $L(\Delta\psi^{2}_{i,j})+R^{1}_{i,j}=0$ for 3 iterations
using initial condition ($\Delta\psi^{2}_{i,j}=0)$

  5. 5.

Mesh level 2: compute $R^{2}_{i,j}=R^{1}_{i,j}+L(\Delta\psi^{2}_{i,j})$

  6. 6.

Mesh level 3: restrict the values of $R^{2}_{i,j}$ from mesh level 2 to the
coarse grid in mesh level 3

  7. 7.

Mesh level 3: solve $L(\Delta\psi^{3}_{i,j})+R^{2}_{i,j}=0$ till convergence
using initial condition ($\Delta\psi^{3}_{i,j}=0$).

  8. 8.

Mesh level 2: interpolate the solution of $\Delta\psi^{3}_{i,j}$ from mesh
level 3 to mesh level 2

  9. 9.

Mesh level 2: solve $L(\Delta\psi^{2}_{i,j})+R^{1}_{i,j}=0$ for 3 iterations
using the new initial condition
$(\Delta\psi^{2}_{i,j})_{new}=\Delta\psi^{2}_{i,j}+\Delta\psi^{3}_{i,j}$

  10. 10.

Mesh level 1: interpolate the solution of $\Delta\psi^{2}_{i,j}$ from mesh
level 2 to mesh level 1

  11. 11.

Mesh level 1: solve $L(\psi^{1}_{i,j})=0$ for 3 iterations using the new
initial condition $(\psi^{1}_{i,j})_{new}=\psi^{1}_{i,j}+\Delta\psi^{2}_{i,j}$

The 11 steps are for one V-cycle. The steps are repeated until the finale
solution is converged.

Figure 7: Multigrid V-cycle

A FORTRAN solver is developed to solve the two-dimensional Laplace equation
using the six iterative methods discussed previously. A flowchart showing the
solver algorithm is described in figure 8. The norm infinity error is defined
as

$\textmd{error}=\textrm{max}\mid\psi^{k+1}_{i,j}-\psi^{k}_{i,j}\mid$ (16)

The stopping criteria is when the error reaches $10^{-9}$.

Figure 8: Solver algorithm

##  5 Results and Discussion

The solver is first verified by solving Laplace equation using two different
set of boundary conditions. The first set of boundary conditions is similar to
that described in the first section. In the second set of boundary conditions,
the top wall has similar conditions as the bottom wall. Figures 9 and 10 show
the contours of $\psi$ obtained from two different boundary conditions. Both
solutions give us confidence that the solver is free from coding errors and
provide correct physical results.

Figure 9: Contours of $\psi$ using BCs described in first section Figure 10:
Contours of $\psi$ using symmetry Bcs

All numerical methods are first compared without over relaxation in order to
give us confidence on the validity of our solver to predict well known trends.
Figure 11 shows the number of iterations and the error of the different
iterative methods employed in the solver. The slowest method is Jacobi which
requires 1194 iterations to reach convergence. GS and SOR with no relaxation
are identical and the number of iterations is 624 iterations, which is
approximately half the number of iterations of Jacobi method. SLORB and SLORA
are much faster than GS and SOR method with only 326 iterations. Although ADI
method is the fastest among these methods with only 326 iterations, it should
be taken into consideration that two calculations are conducted inside a
single cycle of the ADI method. In figure 12, the CPU time for each method is
reported. It can be inferred that Jacobi is the slowest method, GS is the
fastest method and SLOR and ADI methods are approximately equal in terms of
computational time due to the reasons discussed previously.

Figure 11: Error vs. iterations (no over relaxation)

Figure 12: Computational time (no over relaxation)

Now, we are in turn to find the value of the best over relaxation parameter
that would speed up the calculations of the SOR, SLOR and ADI methods. Figure
13 show the effect of the relaxation parameter on the number of iterations. It
can be seen that there exist an optimum value of $1<w<2$ and this value is
different for different iteration methods. The optimum values of $w$ are 1.75,
1.75, 1.25 and 1.3 for SOR, SLORA, SLORB and ADI respectively. Figure 14 shows
the error vs. number of iterations required at the best relaxation parameter
$w$ for each iterative method.

Figure 13: iterations vs. relaxation parameter $w$

Figure 14: Optimum relaxed methods

Figure 15: Summary of Iteration number and CPU time

The number of iterations and CPU times for all methods with optimal relaxation
are summarized in figure 15. As expected, ADI requires less number of
iterations that SLORB (33 vs. 55). However, the total CPU time is almost
identical in both cases. Another important observation is that SLORA is much
slower than SLORB and SOR with optimal over relaxation parameter. Figure 16
compares SLORB and SLORA at their optimal over relaxation parameters. It is
inferred that although $w$ is higher in SLORA, more number of iterations is
required since the solution overshoots in the first 50 iterations. The only
explanation of this observation is low stability of Thomas algorithm while
solving the tri-diagonal matrix when the matrix is not diagonally dominant.

Figure 16: SLORB and SLORA with optimal relaxation parameter

It is also concluded from figure 15 that multigrid method is the most
efficient method among all other iterative methods. The solution converged
after only six V-cycles and the total CPU time is around 1.03 milliseconds
which is fastest relative to all other methods. The initial and final cycles
of the multigid V-cycle method are shown in figures 17 and 18 respectively. It
is clearly seen in cycle 1 that $\Delta\psi$ is in the order of 1. However, at
cycle 6, $\Delta\psi$ is reduced to $10^{-9}$ which is the convergence
criteria defined in this study.

##  6 Conclusion

Laplace equation is an elliptic second order differentioal equation that
describes the behavior of incompressible rotational inviscid flow and heat
transfer in solids. Five point stencil method is a second order finite
difference scheme of the two-dimensional Laplace equation. The solution of the
five stencil finite difference formula is obtained using six different
implicit and explicit iterative methods. Over relaxation is employed in order
to accelerate the iterative procedure. Comparison between different iterative
methods reveal that ADI and SLOR methods are more efficient that other
iterative methods in terms of number of iterations and computational time.
Multigrid V-cycle method is also employed in the current study and the
solution is obtained after only six V-cycles.

Figure 17: Multigrid V-cycle (cycle 1) Figure 18: Multigrid V-cycle (cycle 6)

##  7 Acknowledgment

The author would like to thank Dr. Mingjun Wei for his guidance and
contribution to this work.

## References

  * Pletcher et al. [2012] Pletcher, R. H., Tannehill, J. C., and Anderson, D., _Computational fluid mechanics and heat transfer_ , CRC Press, 2012.