There was no "them", only one point.
And you keep complaining that I addressed them. It was your post. If you did not like it then you should not have posted it.
This is remarkable. You were the one who started banging on about interstellar distances (and then interplanetary for some mysterious reason), not me.
Because distance is the point.
You do not understand that. But I will explain it again.
kilometers 350,000 is about Earth to the Moon
kilometers 200,000,000 is about Earth to Mars
kilometers 39,900,000,000,000 is about Earth to Alpha Centauri
Light travels about 1,000,000 kilometers an hour.
So the exhaust from your example ship will, eventually, disperse beyond your example shield. When that happens, the radiation given off by it will travel at about 1,000,000 kilometers an hour. That means that the math for determining when the enemy will see your ship's silhouette is very simple.
Not in the least, I'm arguing that the heat from the exhaust would have reached negligible levels by the time whatever miniscule amount of it got around the shield, mostly due to the vast majority of it being blocked by the ship and being blasted directly backwards.
I've already given you an example of a laser from Earth to the Moon. Here it is again.
"At the Moon's surface, the beam is about 6.5 kilometers (four miles) wide ..."
https://en.wikipedia.org/wiki/...
So you are claiming that the exhaust from your example ship is MORE tightly focused than a laser is.
The laws of physics disagree with that.
And as another poster pointed out to you, the exhaust isn't nearly as hot as some might imagine.
You don't know how hot I "imagine" it to be. All it has to be is hot enough to be detected. And since the instruments today can (probably) detect leftover radiation from The Big Bang it looks like the laws of physics contradict you again.
And yet again nobody is talking about going from Earth to Mars except yourself.
That is the distance that you quoted. Whether you understood what that meant in actual terms when you quoted it I'm sure that it sounded good to you when you posted it.
What you posted was:
So what you're saying is that from a million plus kilometers away, a ship with a forward profile of maybe a few score meters ...
Now "a million plus kilometers" might sound impressive to someone who does not understand the actual distances in space. But that is just 3x the distance from the Earth to the Moon.
And 1/200th of the distance from the Earth to Mars.
So, yes, detecting an object at that range is easy.
And yet again nobody is talking about going from Earth to Mars except yourself.
I'm pointing out that you do not know what the distances you are quoting mean in the real world.
It the laws of physics.