I am new to photography and I was hoping to go take some pictures of star trails this weekend. I just bought this camera this week and I realized that the highest shutter speed is 15 seconds. Is there any way to keep it open longer? I don't think this camera supports bulb mode.
Based on the specifications at Canon's website, the longest exposure your camera is capable of is 15 seconds. Part of the reason for this is the small size of the sensor, and the resulting small pixel pitch needed to squeeze 12.1MP onto it. All of those pixels that close together generate heat, which can in turn increase the amount of electrical noise that winds up in your images.
One thing you might try is to take a series of photos and combine them to create a star trail photo. Some type of stacking software, such as startrails, is used to combine all the images. Since your camera accepts the optional Canon RS-60E3 cable remote, you could also try a compatible remote timer to automate the series of photos.
Mount the camera on a tripod, using Manual shooting mode select the widest (lowest number) aperture (see page 152 of your manual). Normally you would select the ISO for the number of stars you want to be visible. Fewer bright stars will be visible at lower ISO. Each step higher in ISO will reveal more dim stars in addition to the brighter ones, but will also make the image noisier. Unfortunately, the SX 50 requires a setting of ISO 80 to use exposure times longer than 1.3 seconds (or 1.0 second, depending on which page of the dpreview is correct). You can try it at ISO 80 and see how many, if any, stars are visible in the picture. If you need a higher ISO, you're going to have to take lots of 1.3 second exposures to use with your stacking program. Since you camera supports RAW images, you can use it to have a little more latitude in dealing with noise. With the large number of images you will need to stack, be sure you can do the adjustments on one image and then batch apply them to all of the images before converting to JPEG.
Use manual focus to prevent the camera hunting for focus in between shots, set up the timer to take consecutive shots every 15 seconds, and hit the button. If the camera isn't ready for another shot yet in 15 seconds, change the interval on the timer to 16 seconds. If the ISO forces you to go with shorter shutter speeds, use Continuous LV to enable shooting in burst mode with manual focus (see page 127 of your manual). With this mode you can just lock the button on your cable release down and not worry about setting the timer functions.
You will have to experiment to see how many exposures you can take before noise from your sensor becomes a problem. The cooler the ambient temperature, the longer this should be.
This review on the SX50 says that it can use an RS-60E3 remote switch, although it's not clear from the product description that it works on your camera (and it's not listed under compatibility in the Canon store)
If your camera doesn't have bulb mode, I don't think that will give you bulb functionality anyway.
You would instead need to take multiple frames at 15 seconds, and then use image stacking to put them together.
Not yet. CHDK (the Canon Hack Development Kit) is working on a port for the SX50, but according to their website, it's in very early alpha right now, so if you don't know your way around the camera yet, you don't want to mess with it. However, it will probably be available eventually.