RWS - Lighting a Star with a Flashlight

RWS

Nothing practical, but - if I shine a flashlight into the night sky, pointed at a large but dim star, how many photons per second reaches that star, when the light beam finally arrives?

A cheap 9 LED flashlight produces about 25 lumens of "white" light; probably the combined output of red, green, and blue LEDs. Assume the average energy per visible photon is 2.3 electron volts, about 3.7e-J/photon. A lumen is 683 watts, so the flashlight produces about 37 milliwatts of light, about 1e17 photons per second. The beam is about 50 centimeters wide at 150 centimeters distance, about 0.1 steradian solid angle, so it emits 1e18 photons per steradian per second.

Arcturus is an orange giant, not as intense as a G star like the sun but large. It is 36.7 light years away, D = 3.5e17 meters, and 25 times wider than the sun, a radius r of 1.8e7 kilometers. The star's solid angle from earth is π(r/D)², 8e-21 steradians. 8e-3 photons from my flashlight hits the star every second, about one photon every two minutes ... 36.7 years after I shine my flashlight at it. Surprisingly, more than zero, though impossible to detect compared to the power output of Arcturus.

How about the most powerful pulsed laser, with excellent optics? The uV laser at the National Ignition Facility produces a 500 terawatt, 1.85 megajoule pulse - 37 microseconds. If we assume 10 meter output optics and 351 nm uV (and some magic way to send that up through atmosphere), The beamwidth would be about 1.22 * 351 nm / 10m radians or 4.3e-8 radians, producing a beam solid angle of of 1.5e-15 steradians. 351 nm is 3.5 eV or per photon, so 1.85 megajoules is 3.2e24 photons, or about 2e39 photons per steradian, with 3e24 of the photons reaching the star. 351 nm is 8.5e14 Hz.

Arcturus surface temperature is 4300K, and Boltzmann's constant is 8.62e-5 eV/K, so kT = 0.37 eV. The black body formula is P/hz-m² = 2 ( h \nu) / ( \lambda^2 (e^{h \nu / kT} - 1 ) ) = 2 * 5.6e-19 J / ( (3.51e-7 m)² * 12800 ) = 1.6 nW/m²-Hz , or 1.6e12 W / hz for the entire disk. Our pulse is 37 microseconds, so the bandwidth of a gated detector would be 27 kHz, so the noise power of Arcturus over that band is 4e16 W, but during our gate period it is 1.6e12 joules. The energy of our pulse is a mere 1.8 MJ, and the fraction of that reaching Arcturus is 8e-21/1.5e-15 or 5.5 ppm, about 10 joules. Worse, since the portion of Arcturus facing us is a half-sphere, not a disk, about 60 light seconds in radius, the reflected pulse would be stretched over 60 seconds. We could average many pulses, perhaps, but we might need trillions of years to send them all.

Ah, well, no lidar to Arcturus, even with our best "flashlight".