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Considered as a gedanken experiment are the conditions under which the relativistic Doppler shifting of visible electromagnetic radiation to beyond the human ocular range could reduce the incident radiance of the source, and render a luminous astrophysical body (LAB) invisible to a naked eye. This paper determines the proper distance as a function of relativistic velocity at which a luminous object attains ocular invisibility.

The relativistic blackbody spectrum [^{th} anniversary of Einstein’s general relativity, this note examines, as a gedanken experiment, the specific conditions under which this effect would occur.2

Furthermore, relativistic blackbody radiators will emit spectral radiances which are increased (in the case of approaching) or decreased (in the case of receding), due to temperature inflation and relativistic beaming. By considering in the gedanken experiment the relativistic blackbody spectrum, the proper distances can be determined at which the apparent magnitude of a blackbody radiator is greater (i.e. dimmer) than approximately 6.5 (the threshold of vision for the typical unaided human eye).

Additionally, laboratory tests of the sensitivity of the unassisted human eye are described, and this paper asserts that the Judd & Voss CIE 1978 photopic luminous efficiency function would not be applicable to the situation of LABs due to the much greater luminosity than in the laboratory tests.

The relationship between absolute magnitude, apparent magnitude, and distance to an arbitrary stationary blackbody radiation source has been well established and is given by:

where M is the absolute magnitude of any blackbody radiator, m is its apparent magnitude, and z is the distance to the observer in parsecs. Also, in terms of luminosity,

where M_{o} is the absolute magnitude of a reference star (e.g. the sun), L is the luminosity of the radiation source at an arbitrary distance z, and L_{o} is the absolute luminosity of that source3.

Equating Equations (1) and (2) yields:

Thus, for the sun, M_{o} = 4.83,

Sufficiently high speed relativistic motion of blackbody radiators would clearly Doppler shift the wavelengths of maximum luminosity to beyond the human visual range. Therefore, the lower luminosity wavelengths are Doppler shifted into the visible range, and the overall visible luminosity is reduced.

However, in the case of an approaching blackbody, the radiation is relativistically beamed, and the blackbody temperature is “inflated”. Both of these effects serve to increase the luminosity. For a receding blackbody, relativistic beaming (“expanding”) and temperature “deflation” will have the reverse effect. Therefore, Equation (3) becomes:

where _{o} are the luminosities in the relativistic and rest frames respectively4.

Luminosity is obtained by integrating the spectral radiance over frequency and solid angle:

where ν_{1} and ν_{2} are the mean lower and upper frequencies of ocular visibility. _{o} are the spectral radiances in the relativistic and rest frames respectively, which must be integrated over the appropriate solid angle Ω. The relativistic spectral radiance in frequency space, accounting for Doppler shifting, relativistic beaming, and temperature inflation, was determined by Lee and Cleaver [

where [

and

perature 4-vector, θ is the angle between

The integration of the spectral radiance over all frequencies is straightforward because, with the limits of 0 and ∞, the result is simply π^{4}/15. However, the in-band luminosity requires integration over a finite frequency range. Here, the method of Widger and Woodall is followed [

Letting:

and

Also, letting

Expanding Equation (12) as a difference of integrals:

Evaluating Equation (13), and re-substituting

Expanding the solid angle integration, combining sums, and making use of

(Equation (14)) becomes6:

In the case of approaching the LAB [approximately] directly, a simplification of Equation (15), which cannot be resolved as a closed form function, can be made. Since, θ is very small

Frequently, when evaluating the dΩ integration, the solid angle over which the integration is performed is the solid angle through which the blackbody radiates. However, that is not the case here. The solid angle is that which is subtended by the blackbody

from the vantage point of the observer. Therefore, when

Thus,

Similarly,

Combining Equation (3), in terms of relativistic luminosity, with Equations (18) and (19) yields:

Evaluation of the infinite sums is greatly simplified due, in large part, to the rapid convergence of the series as a result of the

tion (11), is

class star, with a surface temperature of ~50,000 K, and at the lowest frequency of human visibility.

Rν | Number of summation terms (n) |
---|---|

0.1 | 101 |

0.2 | 65 |

0.3 | 50 |

0.4 | 50 |

0.5 | 35 |

0.6 | 30 |

0.7 | 25 |

0.8 | 22 |

0.9 - 1.4 | 20 |

1.5 - 1.9 | 15 |

2.0 - 2.9 | 10 |

3.0 - 3.9 | 8 |

4.0 - 4.9 | 6 |

5.0 - 9.9 | 4 |

10.0 - 24.9 | 3 |

≥25.0 | 1 |

The visibility to the naked eye of astronomical objects has been discussed extensively in the literature [

The efficiency by which photons are used by the retina was accounted for by correcting for the Stiles-Crawford effect of the first (SCE I) and second (SCE II) kind7, photon absorption by the optical media, photopigment absorption of photons, and the photon isomerization efficiency of the photopigment.

For a 22' (diameter), 10 ms, 507 nm monochromatic source, in which, of the ~100 quanta incident upon the retina, 10 to 15 were absorbed by the ~1600 illuminated rods [^{−}^{6} W・m^{−2}・sr^{−}^{1}, and 1.33 × 10^{−3} cd・m^{−2} respectively [^{−4} γ/s, 2.47 × 10^{−9} W・m^{−2}・sr^{−}^{1}, and 7.5 × 10^{−7} cd・m^{−2} respectively.

However, even accounting for the standard observer’s spectral sensitivity by applying the Judd & Voss CIE 1978 photopic luminous efficiency function, these results are difficult to apply to the scenario presented here because of the enormous disparity between the spectral irradiances of the Hallett test sources [

The frequency range of human vision is slightly variable. However, 4.17 × 10^{14} Hz and 7.89 × 10^{14} Hz, which correspond to wavelengths of 720 nm and 380 nm respectively, are acceptable approximations of the limits of human vision, and are in keeping with the wavelengths of 700 nm and 390 nm published by Starr [

In the case of approaching the sun directly (θ = 0), the distance at which the apparent magnitude is 6.5 can be determined from Equation (20), and is shown in

^{9}The angle subtended by the sun at approximately 1 AU.

In order to determine the ocular invisibility curve for an arbitrary velocity vector, the solid angle integration in (15), and correspondingly for the stationary case, must be performed. However, since the solid angle over which the integration must be taken does not significantly exceed ~10 mrad9 (and is considered primarily for angles much smaller), z can be approximated as being constant at each value of θ in the 315 time step iterative scheme, which was used to evaluate the solid angle integral. When Equation (20) is evaluated for the sun,

As expected, and shown in

By making use in this gedanken experiment of the relativistic blackbody spectrum, the velocity profile for the apparent magnitude of a LAB has been determined. Optical invisibility to the unaided eye arises due to the Doppler shifting of the wavelengths of maximum radiance to beyond the limits of human visual sensitivity. Temperature inflation and relativistic beaming can either increase this incident radiance (for an approaching source) or decrease it (for a receding source). By considering the wavelength limits of human vision to be 380 nm and 720 nm, and the limiting magnitude of the unaided human eye to be 6.5, the proper distance versus velocity function for ocular invisibility of relativistic luminous astrophysical bodies has been determined; this profile was determined for the sun.

Whether the physical situations could exist for this effect to be realized is uncertain. As an example, relativistic speeds might be obtainable in the expulsion of a low-mass star from the region of the galactic center as a consequence of a fly-by with the central massive blackhole. Nevertheless, such a star might already appear invisible from earth because of its distance, rather than as a result the relativistic Doppler Effect. The possible realization of this gedanken experiment is an open question.

Lee, J.S. and Cleaver, G.B. (2016) Black Sun: Ocular Invisibility of Relativistic Luminous Astrophysical Bodies. Journal of High Energy Physics, Gravitation and Cosmology, 2, 562-570. http://dx.doi.org/10.4236/jhepgc.2016.24048