From how far away could you see an orbital bombardment?

[RedImperator]

I'm in the process of writing a story in which the viewpoint characters witness an orbital bombardment of Earth from the ground (specifically, Philadelphia, Pennsylvania). The targets are limited to Air Force bases and the like.

The bombardment takes the form of a technobabble lightspeed energy beam that leaves a streak of ionized air behind when it passes through the atmosphere. So it appears to the characters as a thunderbolt extending to the "top of the sky" that fades after a few minutes (the bolt itself is long gone by then, but the atmospheric effects persist). I'm wondering how many of these bolts the characters would see.

How would I calculate how far away an attack would have to be before it's hidden by the horizon? I have them written as seeing three (Dover AFB, McGuire AFB, Willow Grove NAS, the last one close enough for them to feel the ground shock and hear windows rattling), and those are the closest bases to Philadelphia by several hundred miles.

Also, a character in Rochester, NY, at the time of the bombardment, sees two "across the lake" (CFB Trenton and CFB Kingston, both across Lake Ontario from Rochester). Would that be too far away? I didn't think so, especially with nothing but a lake between them and the observer, but I'd like to be sure.

And yes, I know I could just write it, "They saw three flashes nearby, and more further away" and leave it at that, and that's what I'll probably end up doing, but I'm curious about it. Consider this being for my own edification.

[Wicked Pilot]

Dover is the farthest from Philadelphia by only 53 miles, McGuire's only 36, and the Grove is 23. As long as your starships are at least 2000 ft up you'd be able to see the beam in all three locations. I'd recommend though this take place at night under relatively clear skies.

EDIT: And 5800 ft to see from Rochester to Kingston

[Durandal]

Visibility would depend highly on weather conditions and yield, I imagine. The weapon probably doesn't use up a significant portion of its energy doing work to heat the atmosphere around it. If it's really that powerful, it should be at least as visible as a bolt of lightening, and it'd produce a supersonic pressure wave by heating the particles around it until they basically compress and then expand violently.

[drachefly]

I don't see the point of the aftereffect. A well-designed weapon wouldn't spend that much energy doing nothing. Even ordinary lightning scaled up by a couple hundred times wouldn't leave that kind of aftereffect.

As for distance, if it's from orbit, you could see it from thousands of miles away. from high orbit, half the world (if it was bright enough, which it seems like it would be)

[GrandMasterTerwynn]

It's a Wikipedia link, but it does provide the equation necessary to figure out the distance to the horizon. You simply take the elevation of the character, relative to the elevations of the unlucky military installations (or whatever high point might be in the way) and plug it into the equations. If the horizon distance is less than the distance to the targets, then the impact will be below the horizon. Similarly, you could compute the distance from the top of the fireball to the horizon (a sufficiently large release of energy will create a fireball, or a really large column of debris. There is a lot of literature out there on the effects of a nuclear blast, or asteroid impact, which will tell you how high that fireball might get and how big it will be for an energy release of a given magnitude.) If that is tall enough, then your people will be able to see the fireballs. Otherwise, they'll only see flashes of light reflected from clouds and atmospheric dust.

[Sea Skimmer]

This will do the work for you
http://www.boatsafe.com/tools/horizon.htm

[Braedley]

From the way you describe it, it sound's like it'll look a lot like lightning, just on a bigger scale. Lightning also ionizes the air, which is what you see. I doubt the ionizing effect will last as long as want it to, unless the weapon dumps a significant amount of energy into the atmosphere.