I think we can all agree that the crux of this question is in the New Shepard specifications, feel free to offer improvements to these. Here are the ones I arrived at (note: there is no recovery of the booster in this analysis):
Booster Masses:
This reddit thread (that is now 5 yrs old) contains some estimates of tank volume and empty booster mass and I particularly like the comment from poster u/saabstory88:
We know two numbers for sure. The max thrust of the BE-3 is 489kn, and the minimum thrust is 90kn. Assuming that the stage and landing needs to be just a little heavier than the engine can lift at minimum throttle (to be able to hover), then 10,000kg seems like the reasonable base estimate
This other schematic from a less than convincing authority contains a scale:
The top of the tanks seems pretty concrete between the two diagrams (the curvature right below the grid ring unit). It is a bit more ambiguous for the bottom of the tanks. I used a frame from this test-fire video from 2013 and superimposed the exposed engine onto the scale-diagram:
Which seems to place the bottom of the tanks a little higher than the previous diagram. Either way I approximated the tanks (bulk) as a cylinder with a 2.8317 m diameter and a height of 9.3607 m for a volume of 59$m^3$.
This was combined with this answer to Use of different fuels for stages of Saturn V:
while LH2/LOX (also known as hydrolox), has optimal mixture ratios ranging from 4.13 at sea level to 4.83 in vacuum. The STS ran with a ratio of about 6. At 4.12, the bulk density of hydrolox is 0.29g/cm³, at 4.83 its 0.32g/cm³. [emphasis added]
I used 300$\frac{kg}{m^3}$ to arrive at 17700 kg of propellant.
Second Stage: I chose the STAR48BV (TE-M-940-1, pg. 104) because it has thrust vector control capabilities and available specs (and successful history). This configuration looks something like this (proper relative scale):
(the green payload box is ~70 cm x ~50 cm)
Edit: I've updated some of this simulation architecture. Its now 3D and includes a more accurate drag model.
Launch Simulation:
I built a very simple launch simulator to test the vehicle's capabilities. Some 'features' of the sim that are notable (for the inaccuracies they may introduce):
constant drag coefficient- atmospheric model from Braeunig Rocket and Space Technology
- no throttle down for max-Q
non-rotating Earth, meaning the (prograde launching) rocket needs to achieve higher relative speeds (yields more conservative estimates which I don't mind)- constant specific impulses (value used would be somewhere between SL and Vac. value)
- payload fairing jettison at stage separation
- simple pitch program (literally just angle the thrust vector by 'x' until horizontal then maintain altitude)
- 0.1 second time step, Euler integration
- kills sim when permissible orbit reached (regardless of whether stage can actually shut down); semi-major axis > 250 km altitude, eccentricity < 0.1
Here is a table of key inputs to the launch simulator that haven't already been mentioned:
Results: Yes! I was able to simulate putting a 50 kg payload into a 409 km x 168 km x 30.2° orbit. Here is a plot of some key variables, the black vertical lines separate 1st stage burn, coast phase, and 2nd stage burn:
