If Yavin Prime was fired upon, the scattered debris would have destroyed Yavin 4 as well.
Why did they wait to make a direct shot?
If Yavin Prime was fired upon, the scattered debris would have destroyed Yavin 4 as well.
Why did they wait to make a direct shot?
This was addressed in an "Ask the Jedi Council" feature on the StarWars.com website.
According to LucasFilm's (former) Head of Fan Relations/Director of Content Management Steve Sansweet, the Death Star was simply not powerful enough to destroy a gas giant.
Q. In Star Wars, why does the Death Star go around the planet Yavin to blow up the fourth moon when it can simply blow up Yavin first (as it did Alderaan) and then the fourth moon without wasting any time?
A. The Death Star's superlaser is very powerful, but it's not all powerful. Relatively speaking, a terrestrial world of rock and metal like Alderaan is easier to blow up than an immense gas giant like Yavin. The Death Star simply couldn't blow up Yavin, and had to circle the gas giant in order to get to the much smaller moon Yavin 4.
If we begin by assuming that if they could have, they would have, it can be instructive.
How much firepower does it take to destroy a planet the size of Yavin?
We know of one planet destroyed by the Death Star - Alderaan - and we can say that Alderaan is an Earth-type terrestrial planet whereas Yavin is a Jupiter-type gas giant.
If firepower required is a function of planetary mass, then Jupiter, having 317 times the mass of Earth, would require 317 times the firepower to destroy than Earth would.
Likewise Yavin would require a similar magnitude increase in firepower to destroy.
So going back to our original assumption, we conclude that the Empire didn't destroy Yavin because the Death Star didn't have enough firepower to destroy a gas giant that large.
Yavin 4 was being rapidly evacuated at the time of the Death Star's attack.
Debris from Yavin would (likely) strike the moon, but how quickly?
Charles generously did some of the math on the size of Yavin and such, and we could use Kepler's Third Law if we chose, but conveniently, someone has done much of the rest of the math for us.
In short summary, "Yavin IV probably orbits Yavin at about 500,000 km radius". The fastest-moving debris from any man-made explosion was as a result of the Pascal-B nuclear test, a 2,000lb steel plate, which ended up traveling at approximately 66 km/s. Assuming the debris travels at a similar speed, and assuming it would even sufficiently damage Yavin 4, it would end up giving the Rebels a comfortable 126 minutes to evacuate from the moment it was fired.
When the Death Star begins to orbit Yavin, according to the announcing voice, "The moon with the Rebel base will be in range in thirty minutes." By destroying Yavin 4, not Yavin (and as seen in other answers, the ability to do that is not certain) the Empire reduces the amount of time the Rebels have to evacuate by more than an hour.
The Empire was particularly proud of the Death Star and they were power-blinded. They want to see rebels die quickly in one shot. Destroying Yavin would have taken time to destroy Yavin 4.
Plus, the Death Star was near to Yavin and destroying this planet would have caused debris fly to Death Star and damage it. The Death Star didn't have defense shield then. Therefore, it wasn't wise to destroy Yavin. However, this is just speculation.
Plus, if they have destroyed Yavin, you couldn't see the planning of attack and the final battle around Death Star and the movie would have been 30 minutes shorter, wouldn't it? ;-)
The question is, can they destroy it?
Gravitational binding energy U varies as
U ∝ m^2/r ∝ ρ^2r^5 ∝ ρ^{1/3}m^{5/3}
in the uniform density simplification, where ρ is the density, m the mass, and r the radius. Yavin Prime has a diameter 39% larger than Jupiter, but apparently its mass isn't known. Fortunately, we're living in the future! We have a database of exoplanets, and we can search for planets the same size as Yavin Prime. In fact, COROT-5b is almost the exact size of Yavin Prime: 187,000 to 201,000 km compared to Yavin Prime's 198,500 km. It's quite puffy: less than half Jupiter's mass despite being larger. But is this realistic for Yavin Prime?
Yavin Prime's orbital period is 4818 days. If we knew the mass of Yavin, we could use this to determine its distance to Yavin Prime. But Yavin itself is class K, so between 0.45 and 0.8 times the mass of our sun. Kepler's Third Law gives a distance (semimajor axis) of 640 million to 775 million km (lighter = closer) for those figures. By comparison, CoRoT-5b orbits at a mere 7.4 million km, receiving 10,000 times more energy from its star by being 100 times closer. (Actually a bit more, since CoRoT-5 is a bit brighter than Yavin.)
So it's no surprise CoRoT-5 is cooked up to an enormous size. But certainly there's no reason to think Yavin is less dense than CoRoT-5, and it's probably closer to Jupiter which is a similar distance (778 million km) or even a bit denser. So in the likely case it will have gravitational binding energy 1^2 * 1.39^5 = 5.2 times that of Jupiter*, and in the 'worst case' where Yavin Prime is puffy like COROT-5b the GBE is (.217/1.33)^2 * 1.39^5 = 14% that of Jupiter's*. More reasonable would be if Yavin Prime is more like, say, KOI 1421.01 (Uehara et al., "Transiting Planet Candidates Beyond the Snow Line"); with its density Yavin Prime would have a GBE almost three times that of Jupiter's.
But as Jupiter's GBE is thousands of times larger than that of Earth's (the uniform density approximation, which should really not be trusted to scale between a gas giant and a terrestrial planet, gives a factor of 9000). So 2-4 orders of magnitude more than the most we've seen the Death Star produce, after a blast large enough to knock them out for some time as they recharged. So I don't think it's unreasonable to suggest that the Empire didn't destroy Yavin Prime because it was simply beyond their power.
* Using here the uniform density approximation, but only to scale between planets so it's not quite so bad.
My opinion is: What would their reason be?
I mean, how does Grand Moff Tarkin feel about the rebels evacuating the moon or their last ditch attack attempt?
Evacuate? In our moment of triumph? I think you overestimate their chances.
I'd say he's pretty relaxed and confident, all things considered.
He could probably rush it a little, shooting through the outer atmosphere of Yavin Prime when the moon was only just coming to view, but why rush?
As far as he's concerned, victory is assured. That means no need to rush it, no need to focus on little details that, in a heated or uncertain fight, could tip the scales. None of that. He's slowly, deliberately closing in for the kill. This is his moment of glory and he's going to savor every second of it.
Is it safe to assume that Yavin 4 would have been quickly "destroyed" if Yavin were "destroyed?" I'm not sure that it is. I mean, eventually, yes. Yavin 4 would probably end up falling into the local star or whatever. There would certainly be a lot of terrible repercussions for it's ecosystem.
But the question is whether the Rebels would all have been sufficiently killed before they could flee. Killing rebels is what the empire if there for. If Yavin blows up, the local area probably becomes hazardous for navigation, especially for something as large as the Death Star. So, it might be hard to get in close and ensure the rebels are wiped out.
Some pieces of Yavin could, of course, destroy all the rebels on Yavin 4 or perhaps the moon itself. But, they also might not.
Mon Mothma is down there. Leia Organa is down there. The Empire wants to kill those high value targets. What kind of demonstration is it of Imperial power to destroy a gas giant but allow the leaders of the Rebellion to escape. Those would be some mixed signals.
Remember, the Imperials don't know that the Death Star is going to get blown up in five minutes. They are supremely confident. Waiting 5 minutes to line up the death blow makes perfect sense to them.
I think the sudden removal of gravity from the destruction of Yavin Prime could send the Death Star into an unpredictable trajectory.