Hypervelocity Drag
Drag on an accelerating launch loop vehicle with a hemispherical nose
I am not an aeronautical engineer and probably misunderstand the sources. In any case, the numbers are approximate, and should be treated skeptically.
Launch loop capsule drag and heating is acceptable above 80 km altituded. However, incoming debris impactors in decaying orbits will be more unpredictable when the drag is higher, and more difficult to shield or dodge. Perhaps most of them may be intercepted a few orbit earlier, reducing flux at launch loop track level.
Based on Trajectory Optimization for an Apollotype Vehicle under Entry Conditions Encountered During Lunar Return by John W. Young (famous astronaut) and Robert E. Smith Jr., May 1967, NASA TRR258, Langley Research Center.
Equations on Page 5 in FootsecondslugBTU :
1a convective power: ~ ~ \dot Q_c = 20 \rho^{1/2} \left( V \over 1000 \right)^3 Btu/ft^{2}s
1b radiative power: ~ ~ \dot Q_r = 6.1 \rho^{3/2} \left( V \over { 10 000 } \right)^{20} Btu/ft^{2}s
 Equations assume an effective nose radius of 1 foot
Equations from Shock Layer Radiation During Hypervelocity ReEntry by Robert M. Nerem and George H. Stickford, AIAA Entry Technology Conference, CP9, American Institute of Aeronautics and Astronautics, Oct. 1964, pp 158169. (not downloaded or read yet)
Density in slugs/ft^{3}: multiply kg/m^{3} by 1.9403203e3
Power in Btu/ft^{2}s: multiply by 11350.54 to get W/m^{2}
Velocity in ft/s: divide m/s by 0.3048
Metric equations:
Metric 1a convective power: ~ ~ \dot Q_c = 3.53e4 \rho^{1/2} ~ V^3 Watts
Metric 1b radiative power: ~ ~ \dot Q_r = 1.24e69 \rho^{3/2} ~ V^{20} Watts
These are for a 1 foot diameter nose, and scale by {r_n}^{1/2} according to equation 4B4 on page 520 of Part 4B (Entry Heat Transfer) of the SAE Aerospace Applied Thermodynamics Manual. That book cites reference 1, A study of the motion and aerodynamic heating of missiles entering the earth's atmosphere at high supersonic speeds, H. Julian Allen and A. J. Eggers, Jr, NACA TN 4047, 1957. If r_n is in meters, scale by 0.552 {r_n}^{1/2} .
If we scale these for a halfspherical nose, area \pi {r_n}^2 , we get:
Total nose convective power: ~ ~ \dot Q_c = 6.1e4 ( {r_n}^3 ~ \rho )^{1/2} ~ V^3 Watts
Total nose radiative power: ~ ~ \dot Q_r = 2.2e69 ( {r_n} ~ \rho )^{3/2} ~ V^{20} Watts
Effective time:
Assume constant acceleration for the vehicle, v = a t , to a maximum velocity V = a T .
define t_{eff} = {\Large { T \over { n+1 } } } = { \Large { V \over { a ( n+1) } } }
If the drag power \dot Q = k v^n = k a^n t^n , then the time integrated power:
Q=k a^n{\Large {T^{n+1}\over {n+1}}}=k a^n T^n{\Large {T\over{n+1}}} = k V^n t_{eff} = \dot Q_{max} t_{eff}
There will also be additional exit or climbout time for the launch loop added to t_{eff} , TBD. This additional time will be proportionally larger for the radiation fraction, but that will remain small, especially in thinner, higher altitude atmosphere.
The drag losses are much higher; most of the lost energy ends up heating the upper atmosphere (where it radiates efficiently into space, not to the ground). The drag power is P = C_D ~ \rho ~ Area ~ V^3 and the drag loss energy is E = C_D ~ \rho ~ Area ~ V^3 T/4 .
Examples
For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, C_{D} = 2.0 and density at 80, 100, and 120 km:
altitude km 
80 
100 
120 

density kg/m^{3} 
1.85e5 
5.60e7 
2.22e8 

\dot Q_c ~~ kW 
3500 
610 
120 

\dot Q_r ~~ kW 
120 
0.62 
0.005 

exponent 
t_{eff} 

Q_c ~~ MJ 
330 
57 
11 
3 
94 
Q_r ~~ KJ 
2100 
110 
0.09 
20 
18 
Q_{total}~~ MJ 
330 
57 
11 

heat fraction 
1.1e3 
1.9e4 
3.7e5 

drag loss MJ 
14000 
440 
17 

drag fraction 
4.8e2 
1.4e3 
5.7e5 

heat/drag 
0.023 
0.13 
0.65 
Vehicle 
Mass kg 
Diameter 
Length 
Loop 
5000 
2 m 
tbd 
Apollo CM 
5900 
3.9 m 
3.2 m 
Shuttle 
68600 
300 m² surface est. 

Dragon 
4200+3310 
3.7 m 
6.1 m 
Validity?
The Young and Smith paper was for Apollo lunar reentry, and the equations may not generalize to lower drag regions. Exit trajectory must still be computed, keeping in mind that Earth's rotation velocity (470 m/s) should be added to the outgoing orbit velocity. Apollo ballistic parameter 322 kg/m² ± 40%, launch loop 1590 kg/m². A "pill shaped" launch loop capsule may not reenter safely; wings or coneshaped capsule is probably required for manned entry.