Open Access
Table 5.
Comparison of the slopes of the PL and PW relations for BL Her stars of the mathematical form Mλ = alog(P)+b.
| Band | Source | a | b | σ | N | Reference ( ‡ ) | Theoretical/ | (|T|, p(t)) w.r.t. |
|||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Empirical | Set A | Set B | Set C | Set D | |||||||
| Complete set of models (0.5 − 0.8 M⊙) | |||||||||||
| G | Zall (Set A) | −1.76±0.029 | −0.263±0.01 | 0.262 | 3266 | TW | Theoretical | ... | ... | ... | ... |
| G | Zall (Set B) | −1.531±0.031 | −0.261±0.01 | 0.237 | 2260 | TW | Theoretical | (5.352,0.0) | ... | ... | ... |
| G | Zall (Set C) | −1.63±0.032 | −0.171±0.011 | 0.267 | 2632 | TW | Theoretical | (2.996,0.003) | (2.231,0.026) | ... | ... |
| G | Zall (Set D) | −1.49±0.033 | −0.183±0.011 | 0.255 | 2122 | TW | Theoretical | (6.095,0.0) | (0.895,0.371) | (3.049,0.002) | ... |
| G | All Sky | −0.307±0.46 | 15.869±0.151 | 1.57 | 422 | Gaia DR3 | Empirical | (3.152,0.002) | (2.655,0.008) | (2.869,0.004) | (2.565,0.01) |
| G | All Sky (ϖ> 0) | −0.429±0.464 | 15.881±0.151 | 1.566 | 415 | Gaia DR3 | Empirical | (2.863,0.004) | (2.37,0.018) | (2.582,0.01) | (2.281,0.023) |
| G | LMC | −1.237±0.151 | 18.549±0.043 | 0.161 | 61 | Gaia DR3 | Empirical | (3.401,0.001) | (1.907,0.057) | (2.546,0.011) | (1.637,0.102) |
| G | SMC | −2.686±0.721 | 19.022±0.214 | 0.324 | 16 | Gaia DR3 | Empirical | (1.283,0.199) | (1.6,0.11) | (1.463,0.144) | (1.657,0.098) |
| GBP | Zall (Set A) | −1.543±0.033 | −0.077±0.011 | 0.293 | 3266 | TW | Theoretical | ... | ... | ... | ... |
| GBP | Zall (Set B) | −1.296±0.034 | −0.068±0.011 | 0.262 | 2260 | TW | Theoretical | (5.193,0.0) | ... | ... | ... |
| GBP | Zall (Set C) | −1.382±0.036 | 0.034±0.012 | 0.299 | 2632 | TW | Theoretical | (3.33,0.001) | (1.731,0.084) | ... | ... |
| GBP | Zall (Set D) | −1.238±0.037 | 0.024±0.013 | 0.284 | 2122 | TW | Theoretical | (6.178,0.0) | (1.151,0.25) | (2.804,0.005) | ... |
| GBP | All Sky | 0.265±0.593 | 16.504±0.195 | 1.996 | 404 | Gaia DR3 | Empirical | (3.044,0.002) | (2.628,0.009) | (2.772,0.006) | (2.53,0.011) |
| GBP | All Sky (ϖ> 0) | 0.089±0.596 | 16.52±0.194 | 1.982 | 397 | Gaia DR3 | Empirical | (2.734,0.006) | (2.32,0.02) | (2.464,0.014) | (2.222,0.026) |
| GBP | LMC | −0.633±0.266 | 18.648±0.077 | 0.258 | 55 | Gaia DR3 | Empirical | (3.395,0.001) | (2.472,0.013) | (2.79,0.005) | (2.253,0.024) |
| GBP | SMC | −2.866±0.951 | 19.306±0.282 | 0.403 | 14 | Gaia DR3 | Empirical | (1.39,0.165) | (1.65,0.099) | (1.559,0.119) | (1.711,0.087) |
| GRP | Zall (Set A) | −2.015±0.025 | −0.588±0.009 | 0.225 | 3266 | TW | Theoretical | ... | ... | ... | ... |
| GRP | Zall (Set B) | −1.81±0.027 | −0.594±0.009 | 0.209 | 2260 | TW | Theoretical | (5.503,0.0) | ... | ... | ... |
| GRP | Zall (Set C) | −1.909±0.028 | −0.523±0.01 | 0.231 | 2632 | TW | Theoretical | (2.819,0.005) | (2.565,0.01) | ... | ... |
| GRP | Zall (Set D) | −1.779±0.029 | −0.536±0.01 | 0.223 | 2122 | TW | Theoretical | (6.144,0.0) | (0.783,0.434) | (3.271,0.001) | ... |
| GRP | All Sky | −0.692±0.401 | 15.032±0.132 | 1.356 | 408 | Gaia DR3 | Empirical | (3.293,0.001) | (2.782,0.005) | (3.028,0.002) | (2.704,0.007) |
| GRP | All Sky (ϖ> 0) | −0.801±0.405 | 15.043±0.132 | 1.354 | 401 | Gaia DR3 | Empirical | (2.992,0.003) | (2.486,0.013) | (2.729,0.006) | (2.409,0.016) |
| GRP | LMC | −1.769±0.205 | 18.149±0.061 | 0.206 | 55 | Gaia DR3 | Empirical | (1.191,0.234) | (0.198,0.843) | (0.677,0.499) | (0.048,0.961) |
| GRP | SMC | −2.777±0.701 | 18.602±0.208 | 0.297 | 14 | Gaia DR3 | Empirical | (1.086,0.277) | (1.378,0.168) | (1.237,0.216) | (1.422,0.155) |
| W(G, GBP − GRP) | Zall (Set A) | −2.656±0.018 | −1.234±0.006 | 0.159 | 3266 | TW | Theoretical | ... | ... | ... | ... |
| W(G, GBP − GRP) | Zall (Set B) | −2.507±0.021 | −1.261±0.007 | 0.159 | 2260 | TW | Theoretical | (5.432,0.0) | ... | ... | ... |
| W(G, GBP − GRP) | Zall (Set C) | −2.633±0.019 | −1.231±0.007 | 0.163 | 2632 | TW | Theoretical | (0.887,0.375) | (4.412,0.0) | ... | ... |
| W(G, GBP − GRP) | Zall (Set D) | −2.517±0.021 | −1.247±0.007 | 0.165 | 2122 | TW | Theoretical | (4.98,0.0) | (0.341,0.733) | (3.994,0.0) | ... |
| W(G, GBP − GRP) | All Sky | −2.031±0.3 | 12.932±0.098 | 0.982 | 409 | Gaia DR3 | Empirical | (2.08,0.038) | (1.583,0.114) | (2.003,0.045) | (1.616,0.106) |
| W(G, GBP − GRP) | All Sky(ϖ> 0) | −2.044±0.304 | 12.935±0.098 | 0.981 | 402 | Gaia DR3 | Empirical | (2.01,0.045) | (1.519,0.129) | (1.934,0.053) | (1.552,0.121) |
| W(G, GBP − GRP) | LMC | −2.398±0.146 | 17.379±0.044 | 0.144 | 58 | Gaia DR3 | Empirical | (1.754,0.08) | (0.739,0.46) | (1.596,0.111) | (0.807,0.42) |
| W(G, GBP − GRP) | SMC | −2.456±0.439 | 17.662±0.13 | 0.186 | 14 | Gaia DR3 | Empirical | (0.455,0.649) | (0.116,0.908) | (0.403,0.687) | (0.139,0.89) |
| Low-mass models only (0.5 − 0.6M⊙) | |||||||||||
| G | Zall (Set A) | −1.346±0.045 | −0.192±0.015 | 0.222 | 1050 | TW | Theoretical | ... | ... | ... | ... |
| G | Zall (Set B) | −1.223±0.04 | −0.151±0.013 | 0.178 | 707 | TW | Theoretical | (2.049,0.041) | ... | ... | ... |
| G | Zall (Set C) | −1.465±0.048 | −0.014±0.016 | 0.234 | 856 | TW | Theoretical | (1.829,0.068) | (3.894,0.0) | ... | ... |
| G | Zall (Set D) | −1.432±0.046 | −0.007±0.016 | 0.222 | 711 | TW | Theoretical | (1.343,0.179) | (3.42,0.001) | (0.498,0.619) | ... |
| G | All Sky | −0.307±0.46 | 15.869±0.151 | 1.57 | 422 | Gaia DR3 | Empirical | (2.248,0.025) | (1.984,0.048) | (2.504,0.012) | (2.434,0.015) |
| G | All Sky (ϖ> 0) | −0.429±0.464 | 15.881±0.151 | 1.566 | 415 | Gaia DR3 | Empirical | (1.967,0.049) | (1.705,0.088) | (2.221,0.027) | (2.151,0.032) |
| G | LMC | −1.237±0.151 | 18.549±0.043 | 0.161 | 61 | Gaia DR3 | Empirical | (0.692,0.489) | (0.09,0.929) | (1.439,0.15) | (1.235,0.217) |
| G | SMC | −2.686±0.721 | 19.022±0.214 | 0.324 | 16 | Gaia DR3 | Empirical | (1.855,0.064) | (2.026,0.043) | (1.69,0.091) | (1.736,0.083) |
| GBP | Zall (Set A) | −1.074±0.051 | −0.017±0.017 | 0.254 | 1050 | TW | Theoretical | ... | ... | ... | ... |
| GBP | Zall (Set B) | −0.952±0.045 | 0.037±0.015 | 0.202 | 707 | TW | Theoretical | (1.78,0.075) | ... | ... | ... |
| GBP | Zall (Set C) | −1.197±0.055 | 0.194±0.019 | 0.269 | 856 | TW | Theoretical | (1.641,0.101) | (3.438,0.001) | ... | ... |
| GBP | Zall (Set D) | −1.179±0.053 | 0.206±0.018 | 0.254 | 711 | TW | Theoretical | (1.426,0.154) | (3.246,0.001) | (0.234,0.815) | ... |
| GBP | All Sky | 0.265±0.593 | 16.504±0.195 | 1.996 | 404 | Gaia DR3 | Empirical | (2.25,0.025) | (2.046,0.041) | (2.455,0.014) | (2.425,0.015) |
| GBP | All Sky (ϖ> 0) | 0.089±0.596 | 16.52±0.194 | 1.982 | 397 | Gaia DR3 | Empirical | (1.944,0.052) | (1.742,0.082) | (2.149,0.032) | (2.119,0.034) |
| GBP | LMC | −0.633±0.266 | 18.648±0.077 | 0.258 | 55 | Gaia DR3 | Empirical | (1.628,0.104) | (1.182,0.237) | (2.076,0.038) | (2.013,0.044) |
| GBP | SMC | −2.866±0.951 | 19.306±0.282 | 0.403 | 14 | Gaia DR3 | Empirical | (1.882,0.06) | (2.01,0.045) | (1.752,0.08) | (1.771,0.077) |
| GRP | Zall (Set A) | −1.669±0.037 | −0.502±0.012 | 0.182 | 1050 | TW | Theoretical | ... | ... | ... | ... |
| GRP | Zall (Set B) | −1.549±0.033 | −0.475±0.011 | 0.149 | 707 | TW | Theoretical | (2.405,0.016) | ... | ... | ... |
| GRP | Zall (Set C) | −1.768±0.039 | −0.367±0.013 | 0.193 | 856 | TW | Theoretical | (1.845,0.065) | (4.237,0.0) | ... | ... |
| GRP | Zall (Set D) | −1.725±0.038 | −0.364±0.013 | 0.183 | 711 | TW | Theoretical | (1.057,0.29) | (3.451,0.001) | (0.786,0.432) | ... |
| GRP | All Sky | −0.692±0.401 | 15.032±0.132 | 1.356 | 408 | Gaia DR3 | Empirical | (2.426,0.015) | (2.13,0.033) | (2.671,0.008) | (2.565,0.01) |
| GRP | All Sky (ϖ> 0) | −0.801±0.405 | 15.043±0.132 | 1.354 | 401 | Gaia DR3 | Empirical | (2.134,0.033) | (1.841,0.066) | (2.377,0.018) | (2.272,0.023) |
| GRP | LMC | −1.769±0.205 | 18.149±0.061 | 0.206 | 55 | Gaia DR3 | Empirical | (0.48,0.631) | (1.06,0.29) | (0.005,0.996) | (0.211,0.833) |
| GRP | SMC | −2.777±0.701 | 18.602±0.208 | 0.297 | 14 | Gaia DR3 | Empirical | (1.578,0.115) | (1.75,0.081) | (1.437,0.151) | (1.499,0.134) |
| W(G, GBP − GRP) | Zall (Set A) | −2.476±0.022 | −1.113±0.007 | 0.107 | 1050 | TW | Theoretical | ... | ... | ... | ... |
| W(G, GBP − GRP) | Zall (Set B) | −2.358±0.022 | −1.124±0.007 | 0.1 | 707 | TW | Theoretical | (3.799,0.0) | ... | ... | ... |
| W(G, GBP − GRP) | Zall (Set C) | −2.551±0.023 | −1.08±0.008 | 0.113 | 856 | TW | Theoretical | (2.37,0.018) | (6.011,0.0) | ... | ... |
| W(G, GBP − GRP) | Zall (Set D) | −2.47±0.023 | −1.089±0.008 | 0.112 | 711 | TW | Theoretical | (0.207,0.836) | (3.448,0.001) | (2.481,0.013) | ... |
| W(G, GBP − GRP) | All Sky | −2.031±0.3 | 12.932±0.098 | 0.982 | 409 | Gaia DR3 | Empirical | (1.479,0.139) | (1.087,0.277) | (1.728,0.084) | (1.459,0.145) |
| W(G, GBP − GRP) | All Sky (ϖ> 0) | −2.044±0.304 | 12.935±0.098 | 0.981 | 402 | Gaia DR3 | Empirical | (1.417,0.157) | (1.03,0.303) | (1.663,0.097) | (1.397,0.163) |
| W(G, GBP − GRP) | LMC | −2.398±0.146 | 17.379±0.044 | 0.144 | 58 | Gaia DR3 | Empirical | (0.528,0.597) | (0.271,0.787) | (1.035,0.301) | (0.487,0.626) |
| W(G, GBP − GRP) | SMC | −2.456±0.439 | 17.662±0.13 | 0.186 | 14 | Gaia DR3 | Empirical | (0.046,0.964) | (0.223,0.824) | (0.216,0.829) | (0.032,0.975) |
Notes. The theoretical relations are derived for the cases of the complete set of models and for the low mass models. ϖ > 0 indicates the subset of observed BL Her stars in the All Sky region with positive parallaxes from Gaia DR3. N is the total number of stars. |T| represents the observed value of the t-statistic, and p(t) gives the probability of acceptance of the null hypothesis (equal slopes). The bold-faced entries indicate that the null hypothesis of the equivalent PL slopes can be rejected.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.