By combining two answers (18.25km/s dV is a reasonable launch to solar escape and ~18km/s total dV of the Saturn V (before the lunar lander)) means yes, the Saturn V could have without any gravity assists.
But the second post makes it clear they are not including payload, like the Apollo spacecraft, so lets do that math:
| Individual stage | Total vessel |
Stage | Full mass | Dry mass| Full mass | Dry mass | Isp | Δv
------+-----------+---------+-----------+----------+-------+--------------
I | 2,300,000 | 131,000 | 2,900,000 | 731,000 | 263s | 3554.2 m/s
II | 480,000 | 36,000 | 600,000 | 156,000 | 421s | 5561.5 m/s
III | 120,800 | 10,000 | 120,800 | 10,000 | 421s | 8796.2 m/s
===========
17911.9 m/s
* masses are in kg
Adding the 50,000kg payload:
| Individual stage | Total vessel |
Stage | Full mass | Dry mass| Full mass | Dry mass | Isp | Δv
------+-----------+---------+-----------+----------+-------+--------------
I | 2,300,000 | 131,000 | 2,950,000 | 781,000 | 263s | 3427.7 m/s
II | 480,000 | 36,000 | 650,000 | 206,000 | 421s | 4744.2 m/s
III | 120,800 | 10,000 | 170,800 | 60,000 | 421s | 4319.1 m/s
===========
12491.0 m/s
* masses are in kg
Fact Check: this lines up with the published takeoff weights for the Apollo missions
So our Apollo spacecraft has 12.5km/s of initial dV and needs to provide an additional 5.75km/s.
Fact Check: Apollo 17 reached 10.8km/s (35,579.5ft/s) by the end of the S-IVBs 2nd burn.
There are a few points that should be addressed on this number:
This is after gravity/drag losses, which, for the sake of the original solar escape calculation was assumed to be 1.7km/s, which gives us a total dV of 12.5km/s, which is bang on our maximum theoretical dV. But we didn't get all that dV from the rocket:
This is with the benefit from the earth's rotation, about 411m/s at Cape Canaveral and that the Saturn V gravity losses were less than 1.7km/s (1.58km/s) Combined, these give us actual performance of 96% of the theoretical value, assuming we spent all our fuel... but that leads to the next point:
None of the stages were completely expended at their respective engine cutoffs and could have been milked for slightly more dV. The third stage was released with about 2.7% of it's fuel left: 6,431lbs of 239,388lbs at ignition of the first burn. It's also worth noting that that does not translate into 2.7% additional dV, it would generally be more, as this is dead weight when jettisoned. The 1st and 2nd stages had 1.3% and 0.6% remaining fuel on cutoff for Apollo 17... whether this was usable fuel I can't say.
This is where we depart a "normal" mission... we hold onto the LM and just burn for solar escape, so we can't directly compare dV from an Apollo mission any longer.
Stated dVs for the parts of the Apollo spacecraft:
- CSM: 2.8km/s
- Lunar Descent Module: 2.5km/s
- Lunar Ascent Module: 2.22km/s
Total dV: 7.52km/s
Not only is this more than what's needed theoretically and even after a real, nominal Saturn V launch like Apollo 17: 18.25 - 12.5 = 5.75km/s. This is also before any gravity assists.
So presumably, on at least some of the Apollo missions, they could have bucked the flight plan and burned for the stars... assuming they wanted to stuff into the LM and suffocate a short while later.
