Mars has a gravitational binding energy of ~5x10^30 jouleS. If a bomb went off on mars that released that much energy, how much danger would the earth be in. Additionally, would debris striking earth be a major concern?
The only danger for Earth from such an event would be related to bombardment caused by the fragments resulting from it.
However, how big would these fragments be? Well, if you are releasing that much energy in a single shot, I doubt you would be getting anything bigger than dust/pebbles.
Of them, there would be a small fraction on a direct trajectory toward Earth, another fraction which would end up hitting Earth after a few swings around the Sun and other planets and what I suspect would be the largest fraction which will never make it to Earth. Considering that even the parts hitting Earth would not get here in a single go, I don't think we should worry too much.
Life on Earth would be significantly disrupted, but might recover within a few hundred million years.
Assuming the mass of what used to be Mars is spread evenly over the interior of a sphere with the same radius as what used to be its orbit, Earth would sweep out a volume of debris with a mass of 3*10^15 kg each year. This represents a billionfold increase over the amount of space dust that currently lands per year. Or, alternatively, it's three times the mass of the asteroid that killed the dinosaurs - but landing as a continual sandblasting over a year, rather than one big hit.
By colliding with the atmosphere at Earth escape velocity, the Mars dust would heat it up somewhat; an average of 36 watts per square meter over Earth's leading hemisphere, day and night, strongest at sunrise. This is quite a bit less than solar irradiance, which is 1380 W/m^2 when the sun is overhead, but would be noticeable and would increase global temperatures.
Depending on the distribution of particle sizes, quite a few would make it to the surface, causing damage to crops, buildings and vehicles, starting forest fires, and killing people and animals. Death by meteorite would go from a freak occurrence to an everyday event. Life at altitude would carry significant hazards, stratospheric flight would become intensely hazardous, and spaceflight all but impossible; no more satellites for communication or Earth sensing.
The amount of Mars dust landing - 6kg per square meter per year, mainly composed of iron and nickel - would cause significant disruption to ecosystems. Worse, it would oxidize in the atmosphere, removing oxygen; this would not be noticeable at first, but within 1000 years vertebrate life would be in severe difficulty. One large unknown is the extent to which aerosols would form in the troposphere, and the resulting dimming and cooling effect; this could cause a "volcanic winter" style disruption to plant life, again similar to the K-T impact event.
As time went by, impact events would worsen significantly, initially from asteroids liberated from the asteroid belt by the absence of Mars's shepherding influence and by disruption caused by frictional braking from Mars dust, then later by asteroids accreted from Mars dust itself. More impactors would make it through to the Earth's surface, causing localized damage and global aerosol cooling. Terrestrial life would be all but wiped out, but life would continue in the oceans.
Eventually the skies would clear and life could move back onto the land... but it had better hurry, because in a billion years the Sun will become too bright for Earth to remain habitable.
Here is a link to a similar question about what would happen to Earth if Pluto exploded.
In my answer I conclude that:
So exactly how the aliens get rid of Pluto could determine whether nobody on Earth notices it for weeks or months at one extreme, or all life on Earth dies out within a few months on the other extreme.
The mass of Mars is about 0.107 Earth masses, and the mass of Pluto is about 0.00218 Earth masses, so Mars has about 49.08 times the mass of Pluto. The average distance between Pluto and Earth is about 39.482 AU, while the average distnce between Mars and Earth is about 1.523 AU, so Pluto is about 25.923 times as far from Earth.
Thus when the cloud of expanding debris from Mars reaches Earth it should be about 25.923 cubed, or 17,420.306 times as dense as the debris cloud from an exploding Pluto, times 49.08 for the greater mass of Mars, for a total of 854,988.6188 times as dense as the debris cloud of Pluto.
But Mars sometimes is only 0.3727 AU from Earth, making Pluto about 105.935 times farther. So a Martian debris cloud expanded far enough for Earth to pass through it at the closest opposition of Mars would be (105.935 X 105.935 X 105.935 X 49.08), or 58,347,595.95 as dense as a Pluto debris cloud reaching Earth.
So if there is any possibility of a Pluto debris cloud harming Earth there would be a much greater probabiilty of a Mars debris cloud harming Earth.
There is an Endor Holocaust theory, claiming that the explosion of the second Death Star would probably wipe out all life on Endor in a relatively short time.
And someone might also have a Yavin Holocaust Theory, speculating whether all life on the moon of Yavin would have been destroyed by the explosion of the first Death Star.
My answer at this thread:
speculates about the distance between the Death Star and Yavin when it was about to fire on Yavin.
And maybe someone should think about a Death Star Holocaust Theory, calculating whether the first Death Star was too close to Alderaan and whether it would have been destroyed by the wreckage of its planetary victim.