Proving equation for duration of transit

[Erenjaeger]

The lecturer went through part of the proof so i am stuck on finishing it off.
equation: ΔT = PR_*/πα
where P is the period of planet around star
R_* is radius of star
α is orbital radius of planet
what I have done so far...
ω (orbital speed) = v/α also = 2π/p (p being a planetry year)
and linear velocity, v = 2R_* /ΔT
⇒ ΔT = 2R_*/α
so there are obviously some steps to get from 2R_*/α to PR_*/πα but i am not sure how to go about solving it.

 

[haruspex]

v/α also = 2π/p (p being a planetry year)
and linear velocity, v = 2R* /ΔT

Right so far.

⇒ ΔT = 2R*/α

How do you get that?

 

[Erenjaeger]

Right so far.

How do you get that?

that part was given to us, and i can't see how to get there from the previous information

 

[haruspex]

that part was given to us, and i can't see how to get there from the previous information

You did not make it clear what you were given, what you had developed and what was to be shown.
You are trying to arrive at

ΔT = PR*/πα

Yes?

If so, ignore the incorrect equation

ΔT = 2R*/α

and work from

v/α also = 2π/p (p being a planetry year)
and linear velocity, v = 2R* /ΔT

 

[Erenjaeger]

You did not make it clear what you were given, what you had developed and what was to be shown.
You are trying to arrive at

Yes?

If so, ignore the incorrect equation

and work from

yes the equation for duration of the transit is given initially as ΔT = PR_*/πα so should i now work from ΔT = 2R_*/α because I'm not sure how to get to that step

 

[haruspex]

yes the equation for duration of the transit is given initially as ΔT = PR_*/πα so should i now work from ΔT = 2R_*/α because I'm not sure how to get to that step

That does not clarify things for me.
What equation are you trying to get to? Is it ΔT = PR_*/πα?
Where does the equation ΔT = 2R_*/α come from? With variables defined as you specified, it is wrong. It is even dimensionally wrong. It divides one distance by another, which is not going to give a time as a result.
If you throw away that wrong equation and just work with v/α = 2π/p and v = 2R* /ΔT you can get ΔT = 2R_*/α.

(I note that it is rather unusual to use α as a distance. It usually represents an angle or an angular acceleration, but neither of those make sense here either.)

 

[Erenjaeger]

That does not clarify things for me.
What equation are you trying to get to? Is it ΔT = PR_*/πα?
Where does the equation ΔT = 2R_*/α come from? With variables defined as you specified, it is wrong. It is even dimensionally wrong. It divides one distance by another, which is not going to give a time as a result.
If you throw away that wrong equation and just work with v/α = 2π/p and v = 2R* /ΔT you can get ΔT = 2R_*/α.

(I note that it is rather unusual to use α as a distance. It usually represents an angle or an angular acceleration, but neither of those make sense here either.)

im just meant to prove the duration of transit equation, which is given to us as ΔT = PR*/πα the working i gave was what the lecturer had given us to start us off but i am not sure how he got from v = 2R_* /ΔT to then ΔT = 2R_*/α.

 

[haruspex]

but i am not sure how he got from v = 2R* /ΔT to then ΔT = 2R*/α.

Either you copied it down wrongly or the lecturer made a mistake. Clearly it should be ΔT = 2R*/v.

 

[Erenjaeger]

Either you copied it down wrongly or the lecturer made a mistake. Clearly it should be ΔT = 2R*/v.

right yeah that is why i have been stuck i thought that it would obviously go to 2R*/v but in his notes it looked like it said α, ill try and work from that then thanks.

 

[Erenjaeger]

Either you copied it down wrongly or the lecturer made a mistake. Clearly it should be ΔT = 2R*/v.

so now I'm stuck from here actually haha

 

[haruspex]

so now I'm stuck from here actually haha

Well, even ΔT = 2R*/v, though it is right, doesn't seem to be a useful step to take.
You have:
v/α = 2π/P
and
v = 2R* /ΔT
and you want an equation which does not involve v. So use one equation to eliminate v from the other.

 

[Erenjaeger]

Well, even ΔT = 2R*/v, though it is right, doesn't seem to be a useful step to take.
You have:
v/α = 2π/P
and
v = 2R* /ΔT
and you want an equation which does not involve v. So use one equation to eliminate v from the other.

ah okay i think i have it,
v/α = 2π/P
v = 2R* /ΔT
therefore (2R*/ΔT)/α = 2π/p ⇒ (2R*/ΔT)⋅p = 2πα ⇒ 2PR*/ΔT = 2πα ⇒ 2PR* = 2παΔT
and finally..
ΔT = 2PR*/2πα and the 2's cancel so you get back to the original equation ⇒ ΔT = PR*/πα
hows that?

 

[haruspex]

ah okay i think i have it,
v/α = 2π/P
v = 2R* /ΔT
therefore (2R*/ΔT)/α = 2π/p ⇒ (2R*/ΔT)⋅p = 2πα ⇒ 2PR*/ΔT = 2πα ⇒ 2PR* = 2παΔT
and finally..
ΔT = 2PR*/2πα and the 2's cancel so you get back to the original equation ⇒ ΔT = PR*/πα
hows that?

Yes.