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I really enjoyed answering this question. It isn't every day that I get to ponder blowing up a whole solar sytem!

The answer depends on whether you have a smart device which beams its energy at its targets reasonably carefully or if it is a simple explosion, which will waste a lot of energy since most of it will miss the planets altogether and coast off into space. I'll do both calculations, here, because it's fun and easy to do with a spreadsheet.

Now, what does it mean to destory a planet or star? What does it take? What it means is that you've pulled the whole body apart, working against its own force of gravity which tends to hold it together. (I'm ignoring internal forces. In reality, rocky bodies like Earth would be somewhat more work to tear apart because rock doesn't like to come apart. The gas bodies like the Sun and the Jovian planets fit my model well, though.) It turns out that you need an amount of energy equivelent to

| (1) |

where G is the gravitational constant, M is the mass of the body and R is its radius. So all we need is mass and radius for the bodies of the solar system (I'll stick with the Sun and the 9 nominal planets). The following table shows those numbers and the needed energy (under Raw Energy)

Body | Mass | Radius | Distance from the Sun | Raw energy for explosion | Total Sun-centered explosion | |

[grams] | [cm] | [cm] | [ergs] | [ergs] | ||

Sun | 2.00E+33 | 6.96E+10 | 2.30E+48 | 1.14937E+15 | ||

Mercury | 3.30E+26 | 2.44E+08 | 5.791E+12 | 1.79E+37 | 4.03E+46 | |

Venus | 4.87E+27 | 6.05E+08 | 1.0821E+13 | 1.57E+39 | 2.00E+48 | |

Earth | 5.97E+27 | 6.38E+08 | 1.496E+13 | 2.24E+39 | 4.93E+48 | |

Mars | 6.42E+26 | 3.39E+08 | 2.2794E+13 | 4.86E+37 | 8.76E+47 | |

Jupiter | 1.90E+30 | 7.14E+09 | 7.78463E+13 | 2.02E+43 | 9.61E+51 | |

Saturn | 5.68E+29 | 6.03E+09 | 1.42675E+14 | 2.14E+42 | 4.80E+51 | |

Uranus | 8.68E+28 | 2.62E+09 | 2.87107E+14 | 1.15E+41 | 5.53E+51 | |

Neptune | 1.02E+30 | 2.52E+09 | 4.49832E+14 | 1.66E+43 | 2.12E+54 | |

Pluto | 1.27E+25 | 1.14E+08 | 5.90646E+14 | 5.68E+34 | 6.13E+48 | |

Needed Energy [ergs] | 2.30E+48 | 2.12E+54 | ||||

Needed Energy [tons of TNT] | 5.50E+31 | 5.09E+37 | ||||

Needed Mass of Matter/Antimater [grams] | 2.55E+27 | 2.35E+33 |

The total needed energy is therefore 5.50x10^{31} tons of TNT (to
use the units of weapons development). Or, being a Star Trek far too, I
looked at it as a matter/anti-matter bomb problem. That still takes
2.55x10^{27} grams of matter and anti-matter. That's about half of
Earth's mass, or one big bomb.

As I alluded to above, this isn't the whole picture. Most bombs go off and
spread their energy out in all directions equally. This means that most of
the blast energy will miss the planets. Each planet has to intercept the
required amount of energy from the blast over its cross-sectional area (pi
R^{2}) when the initial energy has been spread out of 4 pi
a^{2} surface area (where a is the distance to the Sun). Working
this out, you need 4(a/R)^{2} times more energy to blow up each
planet to account for the wasted energy. This energy is computed in the
right column of the table.

Luckily for our hypothetical space-faring demolition men, we don't have to
add all these energies up. We really only need out explosion to be as
large as the largest energy needed (plus the energy needed to destroy the
Sun, which is relatively negligible). This is because energy wasted by
missing Mercury can hit Venus, Earth, Mars or any of the other planets.
And the same goes for the wasted energy with each other planet (except
Pluto, of course). It turns out that Neptune really sets the amount of
energy we need. (In reality, if we had totalled the last column, the
result would be pretty much the same as the energy needed to destroy only
Neptune.) As you can see, it's 5.09x10^{37} tons of TNT. Or, in
other words, if you took a mass of anti-matter a bit larger than half the
mass of our Sun and added it to the Sun, causing a terrible
matter/anti-matter reaction, it would release enough energy to destroy the
Sun and all of the planets.

A matter/anti-matter reaction is the most efficent explosion we know of today. This doesn't mean that we won't find a more efficient one sometime before Voyager's supposed era (we have nearly 500 years, so there's time). But as you can see, that's still one heck of a weapon by today's standards!

In fairness, I should acknoweldge that I got the planetary data from
*Solar System Dynamics* by Murry and Dermott, one of my all-time
favorite textbooks. You can see the derivation of the graviational binding
energy I used in *Stellar Interiors* by Hansen and Kawaler. I got the
conversion from ergs to tons of TNT from http://www.matter-antimatter.com/energy.htm.
And I used tth, availible at http://hutchinson.belmont.ma.us/tth/
, to convert LaTeX into equation (1).

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