No, if we lose the sun, we are done for. Simple as that.

**1-** We don't know what effect the gravitational slingshot would have on Earth's rotation. Earth is currently rotating at 360° per 24 hours. Which gives it a speed of 1675 km/h ([reference][1]). Any sudden difference in this would be quite catastrophic and would probably end humanity.

**2-** Sunlight is _extremely_ important for human (and other mammals') health. "_According to epidemiologist Robyn Lucas at Australian National University, analysis of lifespan versus disease shows that far more lives worldwide could be lost to diseases caused by lack of sunlight than to those caused by too much, and it is inappropriate to recommend total avoidance of sunlight._" Also: _Multiple sclerosis (MS) is least prevalent in the sunniest regions. Exposure to the ultraviolet-B radiation of sunlight appears to be most important and this may operate via vitamin D synthesis._ ([Wikipedia Article][2]). And of course don't forget that sunlight is required for the synthesis of the vitally important vitamin D. (Also read [this article][3] for a quick idea of how essential sunlight is).

**3-** Remember chilling winter nights? Well, without the sun it is going to be one unending chilling winter night forever. No math required. And this chilling winter night would grip Earth as a whole. With no sunlight, the temperatures would plummet _quickly_. The mean temperature in deep space is barely higher than absolute zero (-273° C) and a cozy mean 24°C of Earth would drop dramatically in the absence of the sun. This is a chart of ocean depth versus temperature, present on wikipedia.

[![enter image description here][4]][4]

Mean ocean depth is 3000 meters and the temperature below 1000 meters is less than 6°C. Furthermore, water is a bad conductor of heath and if moon is shot away from Earth, a layer of ice would quickly form on the tide-less oceans, trapping the heat below from radiating into the environment. And in the presence of a moon (and tides) the motion of top layers of water would quickly radiate away their heat in the chill of deep space. Damned if you do (have a moon) and damned if you don't!

**4-** Did I mention that sunlight exposure is vital for children? Children deprived of essential time under sunlight develop short-sightedness. If we survive for 40 years (extremely, extremely improbable), 70% of the next generation people would have severe cases of myopia and rickets. The rest 30% would be frighteningly ghastly and barely able to feed, let alone actively work.

**5-** We cannot grow crops in the absence of sunlight. No, not with artificial lighting. First of all, how would we establish hundreds of thousands of miles of fertile, underground farmlands when we ourselves would have barely the liveable space? Secondly, how are we going to provide sufficient lighting to this enormous area? The sun provides Earth with 1,575 x 10$^18$ joules per year. This is several times greater than the combined energy consumption of the world and this is how plants get vast amounts of free energy. ([Reference][5]). So no, you cannot hope to grow crops underground in the absence of sun. At all not.

#Conclusion

No, we cannot survive in the long run (more than a few weeks) without the sun.


  [1]: http://www.universetoday.com/26623/how-fast-does-the-earth-rotate/
  [2]: https://en.wikipedia.org/wiki/Health_effects_of_sunlight_exposure
  [3]: http://articles.mercola.com/sites/articles/archive/2012/09/29/sun-exposure-vitamin-d-production-benefits.aspx
  [4]: https://i.sstatic.net/9zJ4k.png
  [5]: https://en.wikipedia.org/wiki/Solar_energy