How do I diagnose/fix a 503 - Service Unavailable Error?

My site PushingSocial.com has been throwing regular 503 Service Unavailable errors. I’ve upgraded the droplet and still get the error. The server is Ubuntu with a LAMP stack

Here’s the latest from the Apache Error log:

[Sun Dec 29 23:45:07.075133 2013] [core:error] [pid 1075:tid 139640860776256] AH00046: child process 1850 still did not exit, sending a SIGKILL [Sun Dec 29 23:45:07.076712 2013] [core:error] [pid 1075:tid 139640860776256] AH00046: child process 2149 still did not exit, sending a SIGKILL [Sun Dec 29 23:45:07.078243 2013] [core:error] [pid 1075:tid 139640860776256] AH00046: child process 2181 still did not exit, sending a SIGKILL [Sun Dec 29 23:45:07.078294 2013] [core:error] [pid 1075:tid 139640860776256] AH00046: child process 2241 still did not exit, sending a SIGKILL [Sun Dec 29 23:45:07.078562 2013] [core:error] [pid 1075:tid 139640860776256] AH00046: child process 2271 still did not exit, sending a SIGKILL [Sun Dec 29 23:45:08.082003 2013] [mpm_event:notice] [pid 1075:tid 139640860776256] AH00491: caught SIGTERM, shutting down [Sun Dec 29 23:45:23.065270 2013] [auth_digest:notice] [pid 1068:tid 140370826737472] AH01757: generating secret for digest authentication … [Sun Dec 29 23:45:24.030813 2013] [ssl:warn] [pid 1068:tid 140370826737472] AH01873: Init: Session Cache is not configured [hint: SSLSessionCache] [Sun Dec 29 23:45:24.030909 2013] [lbmethod_heartbeat:notice] [pid 1068:tid 140370826737472] AH02282: No slotmem from mod_heartmonitor [Sun Dec 29 23:45:24.033948 2013] [mpm_event:notice] [pid 1068:tid 140370826737472] AH00489: Apache/2.4.6 (Unix) OpenSSL/1.0.1 configured – resuming normal operations [Sun Dec 29 23:45:24.033990 2013] [core:notice] [pid 1068:tid 140370826737472] AH00094: Command line: ‘/opt/sp/apache/bin/httpd’ [Mon Dec 30 00:00:04.079698 2013] [auth_digest:notice] [pid 1079:tid 139990628038464] AH01757: generating secret for digest authentication … [Mon Dec 30 00:00:05.051004 2013] [ssl:warn] [pid 1079:tid 139990628038464] AH01873: Init: Session Cache is not configured [hint: SSLSessionCache] [Mon Dec 30 00:00:05.051099 2013] [lbmethod_heartbeat:notice] [pid 1079:tid 139990628038464] AH02282: No slotmem from mod_heartmonitor [Mon Dec 30 00:00:05.053647 2013] [mpm_event:notice] [pid 1079:tid 139990628038464] AH00489: Apache/2.4.6 (Unix) OpenSSL/1.0.1 configured – resuming normal operations [Mon Dec 30 00:00:05.053676 2013] [core:notice] [pid 1079:tid 139990628038464] AH00094: Command line: ‘/opt/sp/apache/bin/httpd’

I appreciate any help you can offer