| Summary: | We highlight here the fact that the distantly observed luminosity of a spherically symmetric compact star radiating thermal radiation isotropically is higher by a factor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></semantics></math></inline-formula> compared to the corresponding flat space-time case, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub></semantics></math></inline-formula> is the surface gravitational redshift of the compact star. In particular, we emphasize that if the thermal radiation is indeed emitted isotropically along the respective normal directions at each point, this factor of increment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></semantics></math></inline-formula> remains unchanged even if the compact object would lie within its <i>photon sphere</i>. Since a canonical neutron star has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo>≈</mo><mn>0.1</mn></mrow></semantics></math></inline-formula>, the actual X-ray luminosity from the neutron star surface could be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∼</mo><mn>20</mn><mo>%</mo></mrow></semantics></math></inline-formula> higher than what would be interpreted by ignoring the general relativistic effects described here. For a static compact object, supported by only isotropic pressure, compactness is limited by the Buchdahl limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo><</mo><mn>2.0</mn></mrow></semantics></math></inline-formula>. However, for compact objects supported by anisotropic pressure, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub></semantics></math></inline-formula> could be even higher (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo><</mo><mn>5.211</mn></mrow></semantics></math></inline-formula>). In addition, in principle, there could be ultra-compact objects having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>z</mi><mi mathvariant="normal">b</mi></msub><mo>≫</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Accordingly, the general relativistic effects described here might be quite important for studies of thermal radiation from some ultra-compact objects.
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